Bi-objective Mathematical Programming Model Research Articles

Decision support for allocating farmed fish to customer orders using a bi-objective optimization model

Aquaculture is an important industry in certain coastal areas. Focusing on farming of salmon and trout, an operational planning problem arises with the goal of allocating supply of fish to demand expressed through customer orders. This paper provides a conceptual model of such a planning problem, defines a corresponding bi-objective mathematical programming model. The problem is novel with respect to the structure of the transportation of fish, and the rules for satisfying customer orders with respect to fish size, quality, certification, and health status. Computational experiments are conducted to gain further insights into the use of the provided model to provide support for planners involved in the operational decision making. Results indicate that the bi-objective optimization model can be useful in situations where supply is insufficient to cover all the demand within a given planning horizon.

Green two-echelon closed and open location-routing problem: application of NSGA-II and MOGWO metaheuristic approaches.

Nowadays organizations outsource transportation of goods or services to reduce cost which leads to a particular type of problem called open location-routing. Also, each logistic organization possesses a limited number of specific vehicles that may not be enough in certain circumstances. This issue indicates the importance of simultaneously considering both open and closed routs. On the other hand, the growing concerns about the detrimental environmental impacts of human activities reveal the necessity of paying attention to environmental issues in logistics. In this study, a bi-objective mathematical programming model is proposed for two-echelon close and open location-routing problem (2E-COLRP) including two echelons of factories, depots and customers to minimize costs and CO2 emissions. The proposed model finds the optimal routs, optimal number of vehicles and facilities as well as the locations of facilities. The augmented epsilon constraint method is used as an exact method to solve the small-sized problems. Due to complexity of model, two metaheuristic algorithms named MOGWO and NSGA-II are utilized to tackle the problems. The efficiency of two aforementioned algorithms is evaluated in terms of several indices considering 22 problem instances with various sizes. The results show that MOGWO performs better than NSGA-II.

Multi-mode project portfolio selection and scheduling in a build-operate-transfer environment

Integrated decision-making about selecting and scheduling a project portfolio can lead to a more desirable performance. This paper considers the problem of selecting a portfolio among a set of possible projects which should be scheduled from vantage point of contractor simultaneously. Each project can be implemented in several modes, each of which has potential cash flows over the time horizon. Therefore, an NPV can be calculated for each project in a particular implementation mode. In this contribution, a structure composed of a fuzzy hybrid multi-criteria method and a fuzzy bi-objective mathematical programming model are presented to deal with quantitative and qualitative dimensions of the project portfolio selection and scheduling problem. To solve the problem, first the qualitative criteria are weighted by the fuzzy AHP, the obtained weights are applied to evaluate the projects in the fuzzy TOPSIS. Then, the fuzzy mathematical model is formulated with two objectives, including maximizing NPV of profit and portfolio score based on assumptions close to real conditions of build-operate-transfer (BOT) contract. According to the assumptions, projects have multi-period implementation and operation phases; there are a penalty and reward system to meet the deadlines. Also, the company is able to attract external capital. The fuzzy model is converted to its crisp counterpart, and then the model is solved using the augmented epsilon constraint method with the data of an Iranian company as a case-study. The results show that considering multiple modes for project implementation has better results in comparison with confining the implementation of each project to only one mode (the mode with the maximum NPV).

Multi-Depot Pickup and Delivery Problem with Resource Sharing

Resource sharing (RS) integrated into the optimization of multi-depot pickup and delivery problem (MDPDP) can greatly reduce the logistics operating cost and required transportation resources by reconfiguring the logistics network. This study formulates and solves an MDPDP with RS (MDPDPRS). First, a bi-objective mathematical programming model that minimizes the logistics cost and the number of vehicles is constructed, in which vehicles are allowed to be used multiple times by one or multiple logistics facilities. Second, a two-stage hybrid algorithm composed of a k-means clustering algorithm, a Clark-Wright (CW) algorithm, and a nondominated sorting genetic algorithm II (NSGA-II) is designed. The k-means algorithm is adopted in the first stage to reallocate customers to logistics facilities according to the Manhattan distance between them, by which the computational complexity of solving the MDPDPRS is reduced. In the second stage, CW and NSGA-II are adopted jointly to optimize the vehicle routes and find the Pareto optimal solutions. CW algorithm is used to select the initial solution, which can increase the speed of finding the optimal solution during NSGA-II. Fast nondominated sorting operator and elite strategy selection operator are utilized to maintain the diversity of solutions in NSGA-II. Third, benchmark tests are conducted to verify the performance and effectiveness of the proposed two-stage hybrid algorithm, and numerical results prove that the proposed methodology outperforms the standard NSGA-II and multi-objective particle swarm optimization algorithm. Finally, optimization results of a real-world logistics network from Chongqing confirm the applicability of the mathematical model and the designed solution algorithm. Solving the MDPDPRS provides a management tool for logistics enterprises to improve resource configuration and optimize logistics operation efficiency.

Designing an integrated municipal solid waste management system using a fuzzy chance-constrained programming model considering economic and environmental aspects under uncertainty

In recent decades, municipal solid waste (MSW) management has become one of the major concerns of human societies. Therefore, an efficient MSW system of collecting, recycling, and disposing is required to be designed, maintained and improved continuously. However, designing a proper waste collection management system by controlling the spread of diseases and environmental pollutants is very costly. To provide an efficient MSW management system, this study proposes an integrated location-allocation problem to plan the operations such as collection, recycling, disposal, and transportation under uncertainty. To deal with the problem uncertainty, a triangular fuzzy number approach is employed and finally, a fuzzy chance-constrained programming model is developed. Moreover, the proposed bi-objective mathematical programming model aims to minimize the total costs and the total volume of pollutant emissions simultaneously. To solve and validate the model, a weighted goal programming (WGP) method is employed and solved by CPLEX solver of GAMS software. Finally, a real case study problem was implemented in Qazvin, Iran. The results showed that, in most cases, increasing the confidence level of the case study problem leads to the improvement of objective functions. Managers can take into account the output of this research provided along with a sensitivity analysis under different conditions.

Sustainable smart waste classification and collection system: A bi-objective modeling and optimization approach

In the context of the “smart city”, information and communication technologies (ICT) have become indispensable in the planning and design of modern municipal solid waste management. Due to more waste varieties, there are urgent calls to implement waste classification worldwide, which promotes resource recycling to achieve sustainable development. In this paper, we present an ICT-based smart waste classification and collection system (SWCCS) that is abstracted as a bi-objective mathematical programming model to optimize the waste collection problem. To implement the proposed SWCCS effectively, we design a novel multi-objective hybrid algorithm based on the whale optimization and genetic algorithms (MOGWOA) with an improved convergence factor and a fast, non-dominated sorting method. A comparison of our algorithm with two classical multi-objective algorithms on generated test instances and on a real-world case shows that the proposed MOGWOA is more effective for optimizing the established model. This paper demonstrates how the ICT-based SWCCS works and how it can help sanitation companies improve waste collection both economically and environmentally.

Transit network design with pollution minimization

A critical step in the design of urban transport networks is the determination of the routes and the frequencies of buses. This situation entails a highly combinatorial optimization problem with a complex computational solution, even for small instances. Several studies have addressed such a situation, minimizing travel times as the main objective. However, the growing trend toward the development of sustainable transport operations requires that the design of the network also considers the emissions of toxic gases that result from combustion, which leads to a new variant of this type of problem, called the pollution transit network design problem. In this paper, the problem is formulated as a biobjective mathematical programming model. Complex problem instances are proposed for this problem, and by using a multi-objective genetic algorithm, we approach the unimodal and bimodal version of the problem by taking into account the elastic demand between buses and cars. By using the proposed mathematical programming model and the genetic algorithm for small and large problem instances, respectively, we show that the generated pollutant emissions are drastically reduced without increasing travel times or costs.

Optimizing a robust bi-objective supply chain network considering environmental aspects: a case study in plastic injection industry

ABSTRACTIn recent years, sustainability has been considered as a growing concern in supply chain network design (SCND). Due to legal requirements, economic profitability, environmental sensitivity, and awareness of customers, sustainable supply chain management has been important for government and industry over the years. The aims of this research are to propose a bi-objective mathematical programming model to integrate the flows of the entire chain to improve performance in conditions of uncertainty associated with the plastic injection industry. The proposed model seeks to minimize total costs including transportation costs, operating costs and fixed costs of setting up facilities and on the other hand, is looking to minimize the environmental emission to increase customer satisfaction. The model proposed based on two scenarios has been solved through the robust optimization approach. Also, to solve the proposed model, some of the multi-objective exact methods such as the weighted sum, weighted goal programming, and LP-metric method are used. Furthermore, a weight sensitivity analysis is applied to find the best set of weights for the objectives. Finally, the results are explained and suggestions for future research are presented.

A robust possibilistic programming model for water allocation problem

Over the past few years, water allocation problem has increasingly spotlighted by governments, researchers and practitioners. As water plays an important role in people’s life and business environment, the problem of water allocation should be considered carefully to properly satisfy demand of water consumers. In the real world applications, problems like water allocation are uncertain owing to long-term planning horizon of such problems. Therefore, employing efficient methods for tackling uncertainty of parameters should be regarded by field researchers. In this regard, this paper proposes a bi-objective mathematical programming model for water distribution network design. The extended model maximizes total profit of water distribution as well as maximizing priority of water transferring among water customer zones. Then, to cope effectively with uncertainty of parameters, a novel robust possibilistic programming method is applied. Then, fuzzy and robust fuzzy programming models are compared against each other and output results confirm superiority and effective performance of the robust fuzzy model in the water allocation problem. Also, output results of the extended model show its accurate performance that results in applicability of the model as a strong planning tool in real world cases.

OPINION-AWARE INFLUENCE MAXIMIZATION: HOW TO MAXIMIZE A FAVORITE OPINION IN A SOCIAL NETWORK?

Influence maximization is a well-known problem in the social network analysis literature which is to find a small subset of seed nodes to maximize the diffusion or spread of information. The main application of this problem in the real-world is in viral marketing. However, the classic influence maximization is disabled to model the real-world viral marketing problem, since the effect of the marketing message content and nodes’ opinions have not been considered. In this paper, a modified version of influence maximization which is named as “opinion-aware influence maximization” (OAIM) problem is proposed to make the model more realistic. In this problem, the main objective is to maximize the spread of a desired opinion, by optimizing the message content, rather than the number of infected nodes, which leads to selection of the best set of seed nodes. A nonlinear bi-objective mathematical programming model is developed to model the considered problem. Some transformation techniques are applied to convert the proposed model to a linear single-objective mathematical programming model. The exact solution of the model in small datasets can be obtained by CPLEX algorithm. For the medium and large-scale datasets, a new genetic algorithm is proposed to cope with the size of the problem. Experimental results on some of the well-known datasets show the efficiency and applicability of the proposed OAIM model. In addition, the proposed genetic algorithm overcomes state-of-the-art algorithms.

Allocation balancing and minimization of the expected costs in a bi-objective stochastic capacitated location problem

ABSTRACTWe consider a stochastic location-allocation problem where optimal locations of facilities among potential locations and optimal allocations of stochastic customers to the facilities are determined. Our two main assumptions are: (1) the customer demands have Bernoulli distributions, and (2) the capacity of a facility for accepting customers is limited so that if the number of allocated customers to the facility is more than its capacity, a shortage will occur. The problem is formulated as a bi-objective mathematical programming model where the total sum of fixed costs of establishment of the facilities and the expected values of servicing and shortage costs and also, the differences of the facility workloads have to be minimized. To solve the proposed model, the augmented ε-constraint method is used. A sample problem is tested and analyzed to show the performance of the proposed model.

Bi-objective location problem with balanced allocation of customers and Bernoulli demands: two solution approaches

A bi-objective stochastic capacitated multi-facility location–allocation problem is presented where the customer demands have Bernoulli distributions. The capacity of a facility for accepting customers is limited so that if the number of allocated customers to the facility is more than its capacity, a shortage will occur. The problem is formulated as a bi-objective mathematical programming model. The first objective is to find optimal locations of facilities among potential locations and optimal allocations of stochastic customers to the facilities so that the total sum of fixed costs of establishment of the facilities and the expected values of servicing and shortage costs is minimized. The second objective is to balance the number of allocated customers to the facilities. To solve small problems, the augmented $$\varepsilon $$ -constraint method is used. Also, two metaheuristic solution approaches, non-dominated sorting genetic algorithm II (NSGA-II) and controlled elitist non-dominated sorting genetic algorithm II (CNSGA-II), are presented for solving large problems. Several sample problems are generated and with various criteria are tested to show the performance of the proposed model and the solution approaches.

A bi-objective mixed-model assembly line sequencing problem considering customer satisfaction and customer buying behaviour

ABSTRACTCustomer satisfaction and focusing on the most profitable customers are the key elements to achieving a sustainable competitive advantage in the current business environment. The mixed-model assembly line (MMAL) is an efficient type of manufacturing system that has been used increasingly over recent years to satisfy diverse customer demands. Product sequencing in an MMAL system has a huge effect on the performance of the firm in meeting customer demand and, therefore, on customer satisfaction. This study aims to address a sequencing problem with respect to recency, frequency and monetary factors. A two-stage framework is developed to achieve this goal. First, the prioritized orders based on the preference ranking organization method for enrichment of evaluations (PROMETHEE) are categorized into high-priority and normal-priority orders. Then, a bi-objective mathematical programming model is developed to determine the optimum sequence of products to minimize the total cost, as well as to maximize levels of customer satisfaction. After validation of the proposed model, owing to the NP-hard nature of this problem, two multi-objective metaheuristics are applied to tackle the second stage of this study. Several test problems are presented to evaluate the performance of these algorithms. The results show the superiority of one solution approach over the other.

Solving a bi-objective mathematical programming model for bloodmobiles location routing problem

Solving a bi-objective mathematical programming model for bloodmobiles location routing problem

Blood supply chain network design considering blood group compatibility under uncertainty

This paper addresses the design of a blood supply chain (SC) network considering blood group compatibility. To this aim, a bi-objective mathematical programming model is developed which minimises the total cost as well as the maximum unsatisfied demand. Due to uncertain nature of some input parameters, two novel robust possibilistic programming models are proposed based on credibility measure. The data of a real case study are then used to illustrate the applicability and performance of the proposed models as well as validating the proposed robust possibilistic programming approach. The obtained results show the superiority of the developed models and significant cost savings compared to current existed blood SC network.

Cell Forming and Cell Balancing of Virtual Cellular Manufacturing Systems with Alternative Processing Routes Using Genetic Algorithm

Cellular manufacturing (CM) is one of the most important subfields in the design of manufacturing systems and as a recently emerged field of study and practice, virtual cellular manufacturing (VCM) inherits the importance from CM. One type of VCM problems is VCM with alternative processing routes from which the route for processing each part should be selected. In this research, a bi-objective mathematical programming model is designed in order to obtain optimal routing of parts, the layout of machines and the assignment of cells to locations and to minimize the production costs and to balance the cell loads. The proposed mathematical model is solved by multi-choice goal programming (MCGP). Since CM models are NP-Hard, a genetic algorithm (GA) is utilized to solve the model for large-sized problem instances and the results obtained from both methods are compared. Finally, a conclusion is made and some visions for future works are offered.Cellular manufacturing (CM) is one of the most important subfields in the design of manufacturing systems and as a recently emerged field of study and practice, virtual cellular manufacturing (VCM) inherits the importance from CM. One type of VCM problems is VCM with alternative processing routes from which the route for processing each part should be selected. In this research, a bi-objective mathematical programming model is designed in order to obtain optimal routing of parts, the layout of machines and the assignment of cells to locations and to minimize the production costs and to balance the cell loads. The proposed mathematical model is solved by multi-choice goal programming (MCGP). Since CM models are NP-Hard, a genetic algorithm (GA) is utilized to solve the model for large-sized problem instances and the results obtained from both methods are compared. Finally, a conclusion is made and some visions for future works are offered.

Pareto-improving transportation network design and ownership regimes

Private provision of public roads signifies co-existence of free, public-tolled and private-tolled roads. This paper investigates the Pareto-improving transportation network design problem under various ownership regimes by allowing joint choice of road pricing and capacity enhancement on free links. The problem of interest is formulated as a bi-objective mathematical programming model that considers the travel cost of road users in each origin-destination pair and the investment return of the whole network. The non-dominated Pareto-improving solutions of toll and/or capacity enhancement schemes are sought for achieving a win-win situation. A sufficient condition is provided for the existence of the non-dominated Pareto-improving schemes and then the properties of those schemes are analyzed. It is found that, under some mild assumptions, the optimal capacity enhancement is uniquely determined by the link flow under any non-dominated Pareto-improving scheme. As a result, the joint road pricing and capacity enhancement problem reduces to a bi-objective second-best road pricing problem. A revenue distribution mechanism with return rate guarantee is proposed to implement the non-dominated Pareto-improving schemes.

Bi-objective vibration damping optimization for congested location–pricing problem

This paper presents a bi-objective mathematical programming model for the restricted facility location problem, under a congestion and pricing policy. Motivated by various applications such as locating server on internet mirror sites and communication networks, this research investigates congested systems with immobile servers and stochastic demand as M/M/m/k queues. For this problem, we consider two simultaneous perspectives; (1) customers who desire to limit waiting time for service and (2) service providers who intend to increase profits. We formulate a bi-objective facility location problem with two objective functions: (i) maximizing total profit of the whole system and (ii) minimizing the sum of waiting time in queues; the model type is mixed-integer nonlinear. Then, a multi-objective optimization algorithm based on vibration theory (so-called multi-objective vibration damping optimization (MOVDO)), is developed to solve the model. Moreover, the Taguchi method is also implemented, using a response metric to tune the parameters. The results are analyzed and compared with a non-dominated sorting genetic algorithm (NSGA-II) as a well-developed multi-objective evolutionary optimization algorithm. Computational results demonstrate the efficiency of the proposed MOVDO to solve large-scale problems.

A bi-objective multi-item capacitated lot-sizing model: Two Pareto-based meta-heuristic algorithms

Lot-sizing problems (LSP) form a class of production planning problems in which available quantities are always considered as decision variables in the production plan. The goal of this paper is to present a multi-item capacitated lot-sizing problem (MICLSP) with setup times, safety stock deficit costs, demand shortage costs – both backorder and lost sale states – and different manners of production. Although a considerable amount of research has concentrated on model development and solution procedures in the terms of single-objective problems in the past decade, to make the model more realistic this paper develops a bi-objective mathematical programming model with two conflicting objectives including: (1) minimizing the total cost considered by the production plans including production costs with different manners of production, inventory costs, safety stock deficit costs, shortage costs and setup costs; (2) minimizing the required storage space. Considering that the proposed model is NP-hard, we propose two novel Pareto-based multi-objective meta-heuristic algorithms called multi-objective vibration damping optimization (MOVDO) and the non-dominated ranking genetic algorithm (NRGA) for the literature on LSP. In order to validate the performance of the proposed MOVDO and NRGA, a non-dominated sorting genetic algorithm (NSGA-II), one of the most common multi-objective meta-heuristic algorithms, is applied. The optimal solutions are also reported to justify the results. Finally, we calibrate both algorithms by robust response surface methodology (RSM); then, the results are analysed on some test problems, both graphically and statistically.

Developing a two-phase model to configure uncertain knowledge-based networks

The purpose of the paper is to develop a bi-objective mathematical programming model to design knowledge-based networks with respect to both configuration costs and suppliers' risks minimisation. The study utilises a mixed-integer linear programming model addressing both the deterministic and robust optimisation approaches. The performance and applicability of the model has been tested by a case study in Iran. The comparison between the deterministic and robust optimisation models indicated that the robust model could efficiently deal with uncertainties associated with costs and demand amount. The given model can be applied for NPD and knowledge-based networks, in which the sustainable collaborations with suppliers is accounted for a critical issue. The potential suppliers can be evaluated with respect to their strength and weakness points in running a steady collaboration. It is required to establish reliable relations between the members of a knowledge-based network to strengthen further interactions, investigated here by a two-phase model.