Bi-objective Model Research Articles

Solving bi-objective uncertain stochastic resource allocation problems by the CVaR-based risk measure and decomposition-based multi-objective evolutionary algorithms

This paper investigates the uncertain stochastic resource allocation problem in which the results of a given allocation of resources are described as probabilities and these probabilities are considered to be uncertain from practical aspects. Here uncertainties are introduced by assuming that these probabilities depend on random parameters which are impacted by various factors. The redundancy allocation problem (RAP) and the multi-stage weapon-target assignment (MWTA) problem are special cases of stochastic resource allocation problems. Bi-objective models for the uncertain RAP and MWTA problem in which the conditional value-at-risk measure is used to control the risk brought by uncertainties are presented in this paper. The bi-objective formulation covers the objectives of minimizing the risk of failure of completing activities and the resulting cost of resources. With the aim of determining referenced Pareto fronts, a linearized formulation and an approximated linear formulation are put forward for RAPs and MWTA problems based on problem-specific characteristics, respectively. Two state-of-the-art decomposition-based multi-objective evolutionary algorithms (i.e., MOEA/D-AWA and DMOEA- $$\varepsilon \hbox {C}$$ ) are used to solve the formulated bi-objective problem. In view of differences between MOEA/D-AWA and DMOEA- $$\varepsilon \hbox {C}$$ , two matching schemes inspired by DMOEA- $$\varepsilon \hbox {C}$$ are proposed and embedded in MOEA/D-AWA. Numerical experiments have been performed on a set of uncertain RAP and MWTA instances. Experimental results demonstrate that DMOEA- $$\varepsilon \hbox {C}$$ outperforms MOEA/D-AWA on the majority of test instances and the superiority of DMOEA- $$\varepsilon \hbox {C}$$ can be ascribed to the $$\varepsilon $$ -constraint framework.

A Bi-Objective Model for Multi-Mode Resource Constrained Scheduling Problem to Optimize the Net Present Value of Costs and Makespan

A Bi-Objective Model for Multi-Mode Resource Constrained Scheduling Problem to Optimize the Net Present Value of Costs and Makespan

Designing a robust demand-differentiated platelet supply chain network under disruption and uncertainty

Managing platelets supply chain network has proved challenging. Besides stochastic demand, the high perishability of platelets and the diversity of their demands make the management more intricate. As a motivation to conduct this paper, we investigate a real-world case study facing a variety of platelet demands to satisfy. Being the first-ever study, we contribute a practical method for the efficient design and planning of a multiple platelet-derived products supply chain network. As the most perishable blood product, platelets need to be maintained fresh. Thus, we suggest a bi-objective model make a tradeoff relationship between the network costs and platelets’ freshness. Further, we account for two realistic features that the products are categorized into three main types with respect to their application and lifetime, and hospitals are prioritized based on their specialty and the population of patients they cover. To cope with the uncertainty and objective multiplicity, we develop a mixed approach. The network robustness under uncertainty is controlled by a robust method and the Pareto solutions of the conflicting objectives are obtained via an interactive approach. Further, we take into account real-world scenarios that the network facilities may face disruptions and utilize a robust scenario-based approach to deal with the disruption scenarios. The results demonstrate that although simultaneous demand fluctuation and disruption increase both logistics costs and delivery time, the proposed model is capable of achieving robust solutions that a little increase in the logistic costs obtains a considerable reduction in the level of relative regret. Further, the network will benefit from a favorable saving in the logistic costs only by a little increase in the storage time.

A Bi-Objective Programming Model for Reliable Supply Chain Network Design under Facility Disruption

Supply chain networks generally are composed of four main entity types: supplier, production centers, distribution centers and demand zones that consist of facilities whose activities involve the transformation of raw material into finished products that are later delivered from the suppliers to the end customers. Supply chain network design as the most important strategic decision in supply chain management plays an important role in the overall environmental and economic performance of the supply chain. The nature and complexity of today’s supply chains network make them vulnerable to various risks. One of the most important risks is disruption risk. Disruptions are costly and can be caused by internal or external sources to the supply chain, thus it is crucial that managers take appropriate measures of responses to reduce its negative effects. A recovery time of disrupted facilities and return it to the normal condition can be an important factor for members of the supply chain. In this paper, a bi-objective model is developed for reliable supply chain network design under facility disruption. To solve this model, we have applied two approaches, i.e., e constraint method as an exact method and non- dominated sorting genetic algorithm (NSGAII) as a meta-heuristic method.

Resilient supplier selection in complex product and its subsystems’ supply chain under uncertainty and risk disruption: A case study for satellite components

Recently, the manufactures of complex product and its subsystems have faced a series of disruptions and troublesome behaviors in supplying goods and items. Likewise, suppliers in this area are more likely to be affected by external risks, in turn eventuating in disturbances. Selecting resilient and expedient suppliers dramatically decreases the delay time and costs and contributes to the competitiveness and development of the companies and organizations in this field. In this regard, this paper aims at proposing a bi-objective robust mathematical model to provide resilience supplier selection and order allocation for complex products and its subsystems in response to uncertainty and disruption risks. In the proposed model, a robust optimization approach is deployed, providing stable decisions for the proposed problem. Also, different resilience strategies including restoring supply from occurred disruptions, fortification of the suppliers, using backup suppliers, and utilizing the extra production capacity for suppliers have been devised to tolerate disruptions. Meanwhile, the augmented e-constraint method is used, ensuring the optimal strong Pareto solutions and preventing the weak ones for the proposed bi-objective model. The evaluation of the effectiveness and desirability of the developed model is explored by discussing a real case study, via which helpful managerial insights are gained.

A bi-objective model for a scheduling problem of unrelated parallel batch processing machines with fuzzy parameters by two fuzzy multi-objective meta-heuristics

This paper considers a bi-objective model for a scheduling problem of unrelated parallel batch processing machines to minimize the makespan and maximum tardiness, simultaneously. Each job has a specific size and the data corresponding to its ready time, due date and processing time-dependent machine are uncertain and determined by trapezoidal fuzzy numbers. Each machine has a specific capacity, in which the number of jobs assigned to each batch on the machine does not violate the machine capacity. The batch processing time, the batch ready time and the batch due date are presented by the longest processing time, the longest ready time and the shortest due date of the jobs that belong to the batch, respectively. To determine the longest and shortest time, the method suggested by Jimenez et al.cite{c18} is used for ranking the fuzzy numbers. A bi-objective fuzzy mixed-integer linear programming model is proposed and solved by two exact methods (i.e., two-phase fuzzy and $epsilon$-constraint) for small-sized problems to obtain a set of Pareto solutions. Because the problem belongs to the class NP-hard, two meta-heuristics, namely fuzzy non-dominated sorting genetic algorithm (FNSGA-II) and fuzzy multi-objective discrete teaching–learning-based optimization (FMODTLBO), are proposed. Then, the comparison of results is illustrated to show their performances. Furthermore, a new representation of the solutions is a matrix with two rows and $N$ columns (i.e., jobs) used to assign the jobs to the batches that processed on the machines.

Optimizing a Bi-Objective Mathematical Model for Minimizing Spraying Time and Drift Proportion

The global agriculture sector faces many challenges in its mission to meet the increasing demand for food and fiber. Climate change, increasing population growth, emergence of crop diseases, damage to crops from rodents and critters, and shrinking farming land in some regions are among these challenges. Application of agrochemicals has proven to be an efficient answer to some of these challenges. However, the impacts of these products on human health and the environment combined with the increased requirement for sustainable farming requires the development of optimal spraying practices that would balance out all interests and concerns. In this paper, a mathematical model is developed to jointly minimize spraying time and drift losses. The obtained bi-objective mixed integer nonlinear programming model is solved for a case study example published in the crop protection literature. Optimal solutions are obtained using the weighted sum method and the epsilon-constraint approach. The results showed that valid and reasonable solutions can be obtained by selecting the appropriate combination of boom height, nozzle spacing, nozzle type, and tractor travel speed. Useful insights are obtained through various computational experiments.

Information-Gap Decision Theory for Robust Security-Constrained Unit Commitment of Joint Renewable Energy and Gridable Vehicles

This paper presents a new framework, utilizing information-gap decision theory (IGDT), for multiobjective robust security-constrained unit commitment of generating units in the presence of wind farms and gridable vehicles. Both the wind power and load demand uncertainties are considered, and modeled using a biobjective model. As the main advantage, the framework enables the system operator to take an appropriate operational decision with respect to the extremity of each uncertainty. The proposed problem is solved using a normal boundary intersection technique. Subsequently, VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR), a decision-making tool, is utilized to choose the best Pareto optimal solution. Finally, the IGDT-based framework presented in this paper is validated using a six-bus test system, the IEEE Reliability Test System with 24 buses, and the IEEE 118-bus system.

Locating key stations of a metro network using bi-objective programming: discrete and continuous demand mode

This study proposes two bi-objective optimization problems for locating key stations of a metro network in both discrete and continuous demand modes. Traditionally, designing a metro network based on optimization techniques consists of two approaches. The first approach locates a number of alignments and their stations simultaneously, while the second approach involves locating key stations, designing a core network, and locating secondary stations. In locating key stations processed by a single objective model, the number of produced and attracted trips to the key stations is maximized. This paper considers a second objective for this stage to maximize the coverage of key stations on origin/destination (OD) trips. A fuzzy goal programming model is established to solve the bi-objective model and provide some Pareto-optimal solutions. The previous single objective model and the proposed model with continuous demand mode are applied to a real network. Results show that the proposed model significantly increases the coverage of key stations on OD trips with only a slight reduction in the number of produced and attracted trips.

Supporting the strategic design of public bicycle sharing systems: The experience of a large Mexican city

Public bike sharing systems (BSS) comprises fleets of bicycles available for shared use at a low cost. BSS are gaining popularity as an environmentally friendly mode of transportation; however, their design entails critical decisions to transportation planners and public authorities. The objective of this study was to design a BSS for a large Mexican city taking into account the available budget, the fulfillment of a desired percentage of the predicted demand, the integration of the BSS with the existent public transportation network and the potential environmental benefit of substituting motorized transportation by cycling. A bi-objective optimization model with two objectives -minimize the total cost of the system’s implementation and maximize the reduction of CO 2 emissions- is formulated and solved to determine the optimal location of bike stations and size of the bicycle fleet required to attain a specific level of service. The analysis of the non-dominated solutions to the problem allows the identification of alternative designs that enhance the BSS’ level of service and its contribution to air quality, provided municipal authorities are willing to make tradeoffs between the two objectives and approve a larger investment.

Integrating production scheduling, maintenance planning and energy controlling for the sustainable manufacturing systems under TOU tariff

Climate change pushes the operation managers to take account of energy-saving issues in their decision-making of production scheduling and maintenance planning (PSMP). We address a PSMP problem for a single machine system under Time-of-Use electricity tariff. We consider two objectives including the makespan that measures the service level and the total energy cost that measures the energy sustainability. Both objectives are considered in a bi-objective mathematical model that is further solved using a novel heuristic algorithm consisting of two layers based on the problem decomposition. The inner layer problem, which is solved by a branch & bound algorithm, is to optimise the decision variables of preventive maintenance and machine’s setup. The outer layer problem, which is solved by a hybrid NSGA-II algorithm, is to optimise the sequence of jobs and the amount of inserted buffer time. The effectiveness and efficiency of the algorithm are demonstrated by a series of numerical experiments. The Pareto frontier can serve as a tool for managers to consider energy cost explicitly in making decisions. It is observed in some scenarios that reducing energy cost will not increase the makespan.

Sustainability in a lot-sizing and scheduling problem with delivery time window and sequence-dependent setup cost consideration

Sustainability in manufacturing operations scheduling is a necessary requirement for companies. Although this subject has attracted much attention, joint decisions of lot-sizing and scheduling with sustainability consideration have not been completely examined in the literature. In this regard, we aim to present a bi-objective model to solve a lot-sizing and scheduling problem simultaneously that would maximize the total profit and minimize CO2 emissions. Also, an augmented ε-constraint method is utilized to transform the model into a single-objective formulation. The proposed model considers Earliness/Tardiness penalties with the delivery time window and sequence-dependent setup costs. Moreover, the model is implemented in a case study of steel company and practical conclusions and managerial insights are reported. Results show that the proposed model is useful for production managers in order to schedule their operations.

Arterial Offset Optimization Considering the Delay and Emission of Platoon: A Case Study in Beijing

The effective setting of offsets between intersections on arterial roads can greatly reduce the travel time of vehicles through intersections. However, coordinated control systems of urban arterial roads often do not achieve the desired effect. On the contrary, they are very likely to increase the traffic congestion on arterial roads, resulting in more delays of the platoon with more exhaust emissions, if the coordinated control system does not have effective settings. Meanwhile, taking into account increasing environmental pollution, measures are needed to solve the conflict between environmental and traffic management. Thus, in order to ensure the smooth flow of urban arterial roads while considering the environment, this paper develops a bi-objective offset optimization model, with reducing delays of the platoon on arterial roads as the primary objective, and reducing exhaust emissions as the secondary objective. The proposed bi-objective model is based on the division of platoon operating modes on arterial roads, and more pollutant types, including NOx, HC, and CO, are considered when measuring environmental impact. Further, the modified hierarchical method, combining the branch and bound approach with the introductions of a relaxation coefficient, is employed to solve the model. By introducing a relaxation coefficient, the modified hierarchical method overcomes the defects of the traditional one. Finally, Xi Dajie Road in Beijing was taken as an example. The results showed that the bi-objective offset optimization model, considering both the delays and emissions of the platoon reduced delays by up to 20% and emissions by 7% compared with the existing timing plan. If compared with the offset optimization model considering delays only, such a model increases delays no more than 3% and reduces emissions by 6%.

Multi-objective biased randomised iterated greedy for robust permutation flow shop scheduling problem under disturbances

Nowadays, scheduling problems under different disruptions are a key to become competitive in the global market of this century. Permutation flow shop scheduling problems are very important as they consider one of the important types of scheduling problems. In this paper, we consider a challenging scheduling problem of a permutation flow shop in the presence of different types of real-time events such as new job arrival and machine breakdown. A multi-objective optimisation model that takes into account multiple performance measures in order to minimise the effect of different real-time events is used in this paper. To solve this problem, we apply the proposed multi-objective model and adapt a predictive-reactive based Biased Randomised Iterated Greedy approach for the problem, which is hybridised a Biased Randomisation process and the Iterated Greedy algorithm. Furthermore, the proposed approach is compared against the predictive-reactive based Particle Swarm Optimisation method for the same problem. Additionally, to show the efficiency of the proposed model, we compare this model by testing the predictive-reactive based BRIG approach to two other models: the bi-objective model that consider only two objectives and the classical single-objective model of minimising the makespan. Further statistical analysis is performed in this study by using an Analysis of Variance measure. The extensive experiments and statistical analysis demonstrate that the proposed multi-objective model is better than the other models in reducing the relative percentage deviation. Additionally, despite their simplicity, the BRIG algorithm is shown to be state-of-the-art method that outperforms the Particle Swarm Optimisation algorithm.

A bi-objective game-theoretic model for collaboration formation between software development firms.

Requirement for formation of collaborations has been on increase for the software development industry, especially for smaller to medium sized firms, due to rapid technological advancements, requirements for diversified skills, ever enhancing demands for innovation and fierce competition. Collaborative product development in an alliance enables the firms to benefit from each other’s diversified skills and the experience as a result of which they can develop products more rapidly and of better quality as well resulting in a higher payoff. Also, the development costs decrease. However, to avoid undesired results, selection of an appropriate partner firm for collaboration is of utmost importance keeping in view the objectives of alliance formation of both the strategic partners. One-way partner selection techniques available in the literature are impractical as they enable a firm to rank potential partners only from its own perspective while ignoring their objectives. This problem is addressed by the two-way partner selection techniques, however, they either ignore the payoff distribution criteria or the proposed criteria is unfair. More importantly, existing techniques consider that firm collaborate only with the objective to enhance their financial payoff which might not always be the case. The fact that collaborating firms may have one but different objectives for collaboration, or, each may have multiple objectives is largely neglected. To address the scenarios in which firms may collaborate due to multiple and possibly different objectives, this work proposes a bi-objective game-theoretic model that enables a firm to select an appropriate partner based on the individual preferences of both on the following two objectives: 1) learning and 2) financial revenue. Moreover, this model calculates the pay-off that each firm should get whether only monetary, only in the form of learning or both. The calculation of payoff share is based on the following parameters: 1) individual goals of collaboration of partner selecting firms on the said two objectives, 2) their level of cost contribution, 3) cooperation ratio and 4) knowledge investment difference. Comprehensive analysis of various scenarios is done for the proposed Nash Bargaining payoff distribution model to find the optimum strategy of collaborating firms for each scenario.

Bi-objective offshore supply vessel planning with costs and persistence objectives

This paper introduces a bi-objective model for the offshore supply vessel planning problem (SVPP) in the oil & gas industry. The SVPP consists of determining a new weekly plan for sailing the platform supply vessels whenever the sailing plan needs revising due to some major events, such as the arrival or removal of drilling rigs, and demand increase or reduction in the current platforms. Apart from remaining cost-efficient, the new weekly plan is required to be persistent, i.e., exhibiting few changes from the previous plan. To achieve this, we propose a bi-objective optimization framework that enables the planners to simultaneously take into account both costs and persistence-related objectives. The framework encompasses a bi-objective SVPP model and a genetic search algorithm adapted from Borthen et al. (2018). We show that the proposed algorithm is able to provide high-quality solutions in reasonable time and has the decision support capability in real offshore operation planning.

Bi-objective Optimization Model for Integrated Preventive Maintenance and Flexible Job-shop Scheduling Problem

This paper develops an integrated model for flexible job-shop scheduling problem with the maintenance activities. Reliability models are used to perform the maintenance activities. This model involves two objectives: minimization of the maximum completion time for flexible job-shop production part and minimization of system unavailability for the PM (preventive maintenance) part. To aim the objectives, two decisions must be taken at the same time: assigning n jobs on m machines in order to minimize the maximum completion time and finding the appropriate times to perform PM activities to minimize the system unavailability. These objectives are obtained with considering dependent machine setup times for operations and release times for jobs. In advance, the maintenance activity numbers and PM intervals are not fixed. Two multi objective optimization methods are compared to find the pareto-optimal front in the flexible job-shop problem case. Promising the obtained results, a benchmark with a large number of test instances is employed.

Bi-objective optimization of the joint replenishment problem in a two-echelon supply chain

In this research, a bi-objective optimisation model for the joint replenishment problem (JRP) of a supply chain comprised of a single supplier and multiple retailers is developed, in which the retailers are assumed to be members of a unique distribution company. The mathematical model minimises the supplier's as well as the retailers' cost subject to some constraints. The constraints are the required storage spaces for any retailer, for any product, and for all the products. The benefit of using the JRP policy is shown based on minimising total cost of supplier and retailers. Since the developed model of the problem is NP-hard, the multi-objective meta-heuristic optimisation algorithm of non-dominated sorting genetic algorithm-II (NSGA-II) is proposed to solve the problem. Besides, since no benchmark is available in the literature to verify and validate the results obtained, a non-dominated ranking genetic algorithm (NRGA) is suggested to solve the problem as well. For further validation, the Pareto fronts are compared to lower and upper bounds obtained using a genetic algorithm employed to solve two single-objective problems separately.

A bi-objective optimization approach for configuring surgical trays with ergonomic risk consideration

For a surgical procedure, the instruments a surgeon has requested are grouped into multiple containers that are referred to as trays. Empirical studies have shown that a large percentage of the instruments that are cleaned and sterilized after a procedure were not even used. In addition, each tray needs to be frequently lifted, which puts perioperative nurses and technicians (PNTs) at risk of developing work-related musculoskeletal disorders (WRMSDs), especially when the trays are overweight. The problem of grouping sterile instruments into trays and determining the tray types for each surgeon and procedure is an NP-hard problem known as the tray optimization problem (TOP). We have developed a bi-objective mathematical model to solve the TOP with the aim of concurrently improving the utilization rate of the instruments and minimizing the tray types required for each procedure. We have established a novel heuristic using an iterated local search (ILS) to quickly and efficiently solve the problem. Computational results over historical data show that the designed ILS is superior to known exact methods. Finally, we have shown to what extent changing the weight limit of trays may have implications on the number of tray types and utilization rate of instruments within trays.

Green parallel machines scheduling problem: A bi-objective model and a heuristic algorithm to obtain Pareto frontier

Sustainability consciousness in manufacturing has become an interesting topic for many researchers in recent years. There is also more concern in many companies about reducing energy consumption in manufacturing. Improving environmental health and safety, production cost saving, access to governmental incentives such as grants and tax credits and also improving the brand image are the most important reasons which is leading many companies to an environmental-friendly production planning. For example, one of the most applicable scheduling problems deals with planning jobs on numbers of parallel machines. In such an application, different machines have different technologies and different speed and power of energy consumption in manufacturing similar jobs. This article introduced a mathematical formulation which models the green parallel machines scheduling problem with total energy consumption and total completion time as objectives. Due to high computational complexity of the proposed model, a heuristic algorithm is developed to obtain the exact Pareto frontier of these two objectives with a polynomial complexity. Numerical experiments are presented to show the efficiency and speed of the proposed algorithm compared to solving the model using the optimisation software directly.