Multi-mode resource constrained project scheduling and contractor selection: Mathematical formulation and metaheuristic algorithms

Contractor selection is a matter of particular attraction for project managers whose aim is to complete projects considering time, cost and quality issues. Traditionally, project scheduling and contractor selection decisions are made separately and sequentially. However, it is usually necessary to satisfy some principles and obligations that impose hard constraints to the problem under consideration. Ignoring this important issue and making project scheduling and contractor selection decisions consecutively may be suboptimal to a holistic view that makes all interrelated decisions in an integrated manner. In this paper, an integrated bi-objective optimization model is proposed to deal with Multi-mode Resource Constrained Project Scheduling Problem (MRCPSP) and Contractor Selection (CS) problem, simultaneously. The objective of the proposed model is to minimize the total costs of the project, and minimize the makespan of the project, simultaneously. To solve the integrated MRCPSP-CS, two multi-objective meta-heuristic algorithms, Non-Dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Particle Swarm Optimization algorithm (MOPSO), are adopted, and 30 test problems of different sizes are solved. The parameter tuning is performed using the Taguchi method. Then, diversification metric (DM), mean ideal distance (MID), quality metric (QM) and number of Pareto solutions (NPS) are used to quantify the performance of meta-heuristic algorithms. Analytic Hierarchy Process (AHP), as a prominent multi-attribute decision-making method, is used to determine the relative importance of performance metrics. Computational results show the superior performance of MOPSO compared to NSGA-II for small-, medium- and large-sized test problems. Moreover, a sensitivity analysis shows that by increasing the number of available contractors, not only the makespan of the project is shortened, but also, the value of NPS in the Pareto front increases, which means that the decision maker(s) can make a wider variety of decisions in a more flexible manner.     

A bi-objective continuous review inventory control model: Pareto-based meta-heuristic algorithms

In this paper, a bi-objective multi-product (r,Q) inventory model in which the inventory level is reviewed continuously is proposed. The aim of this work is to find the optimal value for both order quantity and reorder point through minimizing the total cost and maximizing the service level of the proposed model simultaneously. It is assumed that shortage could occur and unsatisfied demand could be backordered, too. There is a budget limitation and storage space constraint in the model. With regard to complexity of the proposed model, several Pareto-based meta-heuristic approaches such as multi-objective vibration damping optimization (MOVDO), multi-objective imperialist competitive algorithm (MOICA), multi-objective particle swarm optimization (MOPSO), non-dominated ranked genetic algorithm (NRGA), and non-dominated sorting genetic algorithm (NSGA-II) are applied to solve the model. In order to compare the results, several numerical examples are generated and then the algorithms were analyzed statistically and graphically.     

Solving a bi-objective unrelated parallel batch processing machines scheduling problem: A comparison study

Nowadays in competitive markets, production organizations are looking to increase their efficiency and optimize manufacturing operations. In addition, batch processor machines (BPMs) are faster and cheaper to carry out operations; thus the performance of manufacturing systems is increased. This paper studies a production scheduling problem on unrelated parallel BPMs with considering the release time and ready time for jobs as well as batch capacity constraints. In unrelated parallel BPMs, modern machines are used in a production line side by side with older machines that have different purchasing costs; so this factor is introduced as a novel objective to calculate the optimum cost for purchasing various machines due to the budget. Thus, a new bi-objective mathematical model is presented to minimize the makespan (i.e., C_max), tardiness/earliness penalties and the purchasing cost of machines simultaneously. The presented model is first coded and solved by the ε-constraint‌ method. Because of the complexity of the NP-hard problem, exact methods are not able to optimally solve large-sized problems in a reasonable time. Therefore, we propose a multi-objective harmony search (MOHS) algorithm. the results are compared with the multi-objective particle swarm optimization (MOPSO), non-dominated sorting genetic algorithm (NSGA-II), and multi-objective ant colony optimization algorithm (MOACO). To tune their parameters, the Taguchi method is used. The results are compared by five metrics that show the effectiveness of the proposed MOHS algorithm compared with the MOPSO, NSGA-II and MOACO. At last, the sensitivity of the model is analyzed on new parameters and impacts of each parameter are illustrated on bi- objective functions.     

Hybrid NSGA-II for an imperfect production system considering product quality and returns under two warranty policies

This study presents a new bi-objective mathematical model for an imperfect production system (IPS) that accounts for product returns and quality levels under two different warranty policies. We considered uncertain customer demand as stochastic behavior and product price as a function of return compensation, product quality level, and warranty-period length. We simultaneously looked at the pro rata warranty and free replacement/repair warranty policies and assumed that customers can choose the desired policy. A return policy for an online distributor was also included and two objective functions were used to address the problem. The first objective function maximized the total expected revenue, and the second objective function maximized customer satisfaction. The non-dominated sorting genetic algorithm (NSGA-II) and two others, the basic bee metaheuristic and teaching-learning-based optimization algorithms, were used to generate the initial solution for use in the NSGA-II algorithm. The results from the hybrid algorithms revealed that the proposed method improves the performance of the NSGA-II algorithm. Finally, several specific sensitivity analyses were conducted to determine the effects of the problem parameters.     

Two new meta-heuristics for a bi-objective supply chain scheduling problem in flow-shop environment

In this study, an integrated multi-objective production-distribution flow-shop scheduling problem will be taken into consideration with respect to two objective functions. The first objective function aims to minimize total weighted tardiness and make-span and the second objective function aims to minimize the summation of total weighted earliness, total weighted number of tardy jobs, inventory costs and total delivery costs. Firstly, a mathematical model is proposed for this problem. After that, two new meta-heuristic algorithms are developed in order to solve the problem. The first algorithm (HCMOPSO), is a multi-objective particle swarm optimization combined with a heuristic mutation operator, Gaussian membership function and a chaotic sequence and the second algorithm (HBNSGA-II), is a non-dominated sorting genetic algorithm II with a heuristic criterion for generation of initial population and a heuristic crossover operator. The proposed HCMOPSO and HBNSGA-II are tested and compared with a Non-dominated Sorting Genetic Algorithm II (NSGA-II), a Multi-Objective Particle Swarm Optimization (MOPSO) and two state-of-the-art algorithms from recent researches, by means of several comparing criteria. The computational experiments demonstrate the outperformance of the proposed HCMOPSO and HBNSGA-II.     

Extraction of decision alternatives in construction management projects: Application and adaptation of NSGA-II and MOPSO

The time-cost trade-off problem is a known bi-objective problem in the field of project management. Recently, a new parameter, the quality of the project has been added to previously considered time and cost parameters. The main specification of the time-cost trade-off problem is discretization of the decision space to limited and accountable decision variables. In this situation the efficiency of the traditional methods decrease and applying of the evolutionary algorithms is necessary. In this paper, two evolutionary algorithms that originally search the decision space in a continuous manner including: (1) multi-objective particle swarm optimization (MOPSO) and (2) nondominated sorting genetic algorithm (NSGA)-II, are considered as the optimization tools to solve two construction project management problems. These problems are both in discrete domain including two or tree objectives, separately. In this regard, some procedures has been suggested and then applied to adopt both algorithms capable in solving the problems in a discrete domain. Results show the advantages and effectiveness of the used procedures in reporting the optimal Pareto for the optimization problems. Moreover, the NSGA-II is more successful in determining optimal alternatives in both time-cost trade-off (TCTO) and time-cost-quality trade-off (TCQTO) problems than the MOPSO algorithm.     

A bi-objective model to optimize reliability and cost of system with a choice of redundancy strategies

Reliability problems are an important type of optimization problems that are motivated by different needs of real-world applications such as telecommunication systems, transformation systems, and electrical systems, so on. This paper studies a special type of these problems which is called redundancy allocation problem (RAP) and develops a bi-objective RAP (BORAP). The model includes non-repairable series–parallel systems in which the redundancy strategy is considered as a decision variable for individual subsystems. The objective functions of the model are (1) maximizing system reliability and (2) minimizing the system cost. Meanwhile, subject to system-level constraint, the best redundancy strategy among active or cold-standby, component type, and the redundancy level for each subsystem should be determined. To have a more practical model, we have also considered non-constant component hazard functions and imperfect switching of cold-standby redundant component. To solve the model, since RAP belong to the NP-hard class of the optimization problems, two effective multi-objective metaheuristic algorithms named non-dominated sorting genetic algorithms (NSGA-II) and multi-objective particle swarm optimization (MOPSO) are proposed. Finally, the performance of the algorithms is analyzed on a typical case and conclusions are demonstrated.     

A multi objective volleyball premier league algorithm for green scheduling identical parallel machines with splitting jobs

Parallel machine scheduling is one of the most common studied problems in recent years, however, this classic optimization problem has to achieve two conflicting objectives, i.e. minimizing the total tardiness and minimizing the total wastes, if the scheduling is done in the context of plastic injection industry where jobs are splitting and molds are important constraints. This paper proposes a mathematical model for scheduling parallel machines with splitting jobs and resource constraints. Two minimization objectives - the total tardiness and the number of waste - are considered, simultaneously. The obtained model is a bi-objective integer linear programming model that is shown to be of NP-hard class optimization problems. In this paper, a novel Multi-Objective Volleyball Premier League (MOVPL) algorithm is presented for solving the aforementioned problem. This algorithm uses the crowding distance concept used in NSGA-II as an extension of the Volleyball Premier League (VPL) that we recently introduced. Furthermore, the results are compared with six multi-objective metaheuristic algorithms of MOPSO, NSGA-II, MOGWO, MOALO, MOEA/D, and SPEA2. Using five standard metrics and ten test problems, the performance of the Pareto-based algorithms was investigated. The results demonstrate that in general, the proposed algorithm has supremacy than the other four algorithms.

     

Minimization of exceptional elements and voids in the cell formation problem using a multi-objective genetic algorithm

Cell formation problem is the main issue in designing cellular manufacturing systems. The most important objective in the cell formation problem is to minimize the number of exceptional elements which helps to reduce the number of intercellular movements. Another important but rarely used objective function is to minimize the number of voids inside of the machine cells. This objective function is considered in order to increase the utilization of the machines. We present a bi-objective mathematical model to simultaneously minimize the number of exceptional elements and the number of voids in the part machine incidence matrix. An ε-constraint method is then applied to solve the model and to generate the efficient solutions. Because of the NP-hardness of the model, the optimal algorithms can not be used in large-scale problems and therefore, we have also developed a bi-objective genetic algorithm. Some numerical examples are considered to illustrate the performance of the model and the effectiveness of the solution algorithms. The results demonstrate that in comparison with the ε-constraint method, the proposed genetic algorithm can obtain efficient solution in a reasonable run time.     

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 soft-computing Pareto-based meta-heuristic algorithm for a multi-objective multi-server facility location problem

In this paper, a novel multi-objective location model within multi-server queuing framework is proposed, in which facilities behave as M/M/m queues. In the developed model of the problem, the constraints of selecting the nearest-facility along with the service level restriction are considered to bring the model closer to reality. Three objective functions are also considered including minimizing (I) sum of the aggregate travel and waiting times, (II) maximum idle time of all facilities, and (III) the budget required to cover the costs of establishing the selected facilities plus server staffing costs. Since the developed model of the problem is of an NP-hard type and inexact solutions are more probable to be obtained, soft computing techniques, specifically evolutionary computations, are generally used to cope with the lack of precision. From different terms of evolutionary computations, this paper proposes a Pareto-based meta-heuristic algorithm called multi-objective harmony search (MOHS) to solve the problem. To validate the results obtained, two popular algorithms including non-dominated sorting genetic algorithm (NSGA-II) and non-dominated ranking genetic algorithm (NRGA) are utilized as well. In order to demonstrate the proposed methodology and to compare the performances in terms of Pareto-based solution measures, the Taguchi approach is first utilized to tune the parameters of the proposed algorithms, where a new response metric named multi-objective coefficient of variation (MOCV) is introduced. Then, the results of implementing the algorithms on some test problems show that the proposed MOHS outperforms the other two algorithms in terms of computational time.     

A multi-objective model for facility location–allocation problem with immobile servers within queuing framework

This research investigates a practical bi-objective model for the facility location–allocation (BOFLA) problem with immobile servers and stochastic demand within the M/M/1/K queue system. The first goal of the research is to develop a mathematical model in which customers and service providers are considered as perspectives. The objectives of the developed model are minimization of the total cost of server provider and minimization of the total time of customers. This model has different real world applications, including locating bank automated teller machines (ATMs), different types of vendor machines, etc. For solving the model, two popular multi-objective evolutionary algorithms (MOEA) of the literature are implemented. The first algorithm is non-dominated sorted genetic algorithm (NSGA-II) and the second one is non-dominated ranked genetic algorithm (NRGA). Moreover, to illustrate the effectiveness of the proposed algorithms, some numerical examples are presented and analyzed statistically. The results indicate that the proposed algorithms provide an effective means to solve the problems.     

A multi-objective model for discovering high-quality knowledge based on data quality and prior knowledge

Discovering knowledge from data means finding useful patterns in data, this process has increased the opportunity and challenge for businesses in the big data era. Meanwhile, improving the quality of the discovered knowledge is important for making correct decisions in an unpredictable environment. Various models have been developed in the past; however, few used both data quality and prior knowledge to control the quality of the discovery processes and results. In this paper, a multi-objective model of knowledge discovery in databases is developed, which aids the discovery process by utilizing prior process knowledge and different measures of data quality. To illustrate the model, association rule mining is considered and formulated as a multi-objective problem that takes into account data quality measures and prior process knowledge instead of a single objective problem. Measures such as confidence, support, comprehensibility and interestingness are used. A Pareto-based integrated multi-objective Artificial Bee Colony (IMOABC) algorithm is developed to solve the problem. Using well-known and publicly available databases, experiments are carried out to compare the performance of IMOABC with NSGA-II, MOPSO and Apriori algorithms, respectively. The computational results show that IMOABC outperforms NSGA-II, MOPSO and Apriori on different measures and it could be easily customized or tailored to be in line with user requirements and still generates high-quality association rules.     

A bi-objective possibilistic programming model for open shop scheduling problems with sequence-dependent setup times, fuzzy processing times, and fuzzy due dates

We are concerned with an open shop scheduling problem having sequence-dependent setup times. A novel bi-objective possibilistic mixed-integer linear programming model is presented. Sequence-dependent setup times, fuzzy processing times and fuzzy due dates with triangular possibility distributions are the main constraints of this model. An open shop scheduling problem with these considerations is close to the real production scheduling conditions. The objective functions are to minimize total weighted tardiness and total weighted completion times. To solve small-sized instances for Pareto-optimal solutions, an interactive fuzzy multi-objective decision making (FMODM) approach, called TH method proposed by Torabi and Hassini, is applied. Using this method, an equivalent auxiliary single-objective crisp model is obtained and solved optimally by the Lingo software. For medium to large size examples, a multi-objective particle swarm optimization (MOPSO) algorithm is proposed. This algorithm consists of a decoding procedure using a permutation list to reduce the search area in the solution space. Also, a local search algorithm is applied to generate good initial particle positions. Finally, to evaluate the effectiveness of the MOPSO algorithm, the results are compared with the ones obtained by the well-known SPEA-II, using design of experiments (DOE) based on some performance metrics.     

A hybrid vendor managed inventory and redundancy allocation optimization problem in supply chain management: An NSGA-II with tuned parameters

In this research, a bi-objective vendor managed inventory model in a supply chain with one vendor (producer) and several retailers is developed, in which determination of the optimal numbers of different machines that work in series to produce a single item is considered. While the demand rates of the retailers are deterministic and known, the constraints are the total budget, required storage space, vendor's total replenishment frequencies, and average inventory. In addition to production and holding costs of the vendor along with the ordering and holding costs of the retailers, the transportation cost of delivering the item to the retailers is also considered in the total chain cost. The aim is to find the order size, the replenishment frequency of the retailers, the optimal traveling tour from the vendor to retailers, and the number of machines so as the total chain cost is minimized while the system reliability of producing the item is maximized. Since the developed model of the problem is NP-hard, the multi-objective meta-heuristic optimization 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. The parameters of both algorithms are first calibrated using the Taguchi approach. Then, the performances of the two algorithms are compared in terms of some multi-objective performance measures. Moreover, a local searcher, named simulated annealing (SA), is used to improve NSGA-II. 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.     

Two parameter tuned multi-objective evolutionary algorithms for a bi-objective vendor managed inventory model with trapezoidal fuzzy demand

This paper presents a bi-objective vendor managed inventory (BOVMI) model for a supply chain problem with a single vendor and multiple retailers, in which the demand is fuzzy and the vendor manages the retailers’ inventory in a central warehouse. The vendor confronts two constraints: number of orders and available budget. In this model, the fuzzy demand is formulated using trapezoidal fuzzy number (TrFN) where the centroid defuzzification method is employed to defuzzify fuzzy output functions. Minimizing both the total inventory cost and the warehouse space are the two objectives of the model. Since the proposed model is formulated into a bi-objective integer nonlinear programming (INLP) problem, the multi-objective evolutionary algorithm (MOEA) of non-dominated sorting genetic algorithm-II (NSGA-II) is developed to find Pareto front solutions. Besides, since there is no benchmark available in the literature to validate the solutions obtained, another MOEA, namely the non-dominated ranking genetic algorithms (NRGA), is developed to solve the problem as well. To improve the performances of both algorithms, their parameters are calibrated using the Taguchi method. Finally, conclusions are made and future research works are recommended.     

An Integer Linear Programming approach to the single and bi-objective Next Release Problem

ContextThe Next Release Problem involves determining the set of requirements to implement in the next release of a software project. When the problem was first formulated in 2001, Integer Linear Programming, an exact method, was found to be impractical because of large execution times. Since then, the problem has mainly been addressed by employing metaheuristic techniques.ObjectiveIn this paper, we investigate if the single-objective and bi-objective Next Release Problem can be solved exactly and how to better approximate the results when exact resolution is costly.MethodsWe revisit Integer Linear Programming for the single-objective version of the problem. In addition, we integrate it within the Epsilon-constraint method to address the bi-objective problem. We also investigate how the Pareto front of the bi-objective problem can be approximated through an anytime deterministic Integer Linear Programming-based algorithm when results are required within strict runtime constraints. Comparisons are carried out against NSGA-II. Experiments are performed on a combination of synthetic and real-world datasets.FindingsWe show that a modern Integer Linear Programming solver is now a viable method for this problem. Large single objective instances and small bi-objective instances can be solved exactly very quickly. On large bi-objective instances, execution times can be significant when calculating the complete Pareto front. However, good approximations can be found effectively.ConclusionThis study suggests that (1) approximation algorithms can be discarded in favor of the exact method for the single-objective instances and small bi-objective instances, (2) the Integer Linear Programming-based approximate algorithm outperforms the NSGA-II genetic approach on large bi-objective instances, and (3) the run times for both methods are low enough to be used in real-world situations.     

A novel multi-objective meta-heuristic model for solving cross-docking scheduling problems

Cross-docking is a material handling and distribution technique in which products are transferred directly from the receiving dock to the shipping dock, reducing the need for a warehouse or distribution center. This process minimizes the storage and order-picking functions in a warehouse. In this paper, we consider cross-docking in a supply chain and propose a multi-objective mathematical model for minimizing the make-span, transportation cost and the number of truck trips in the supply chain. The proposed model allows a truck to travel from a supplier to the cross-dock facility and from the supplier directly to the customers. We propose two meta-heuristic algorithms, the non-dominated sorting genetic algorithm (NSGA-II) and the multi-objective particle swarm optimization (MOPSO), to solve the multi-objective mathematical model. We demonstrate the applicability of the proposed method and exhibit the efficacy of the procedure with a numerical example. The numerical results show the relative superiority of the NSGA-II method over the MOPSO method.     

基于多目标飞蛾算法的电力系统无功优化研究

鉴于电力需求的日益增长与传统无功优化方法的桎梏,如何更加合理有效地解决电力系统的无功优化问题逐渐成为了研究的热点。提出一种多目标飞蛾扑火算法来解决电力系统多目标无功优化的问题,算法引入固定大小的外部储存机制、自适应的网格和筛选机制来有效存储和提升无功优化问题的帕累托最优解集,算法采用CEC2009标准多目标测试函数来进行仿真实验,并与两种经典算法进行性能的对比分析。此外,在电力系统IEEE 30节点上将该算法与MOPSO,NGSGA-II算法的求解结果进行比较分析的结果表明,多目标飞蛾算法具有良好的性能,并在解决电力系统多目标无功优化问题上具有良好的潜力。     

A new optimization approach for nozzle selection and component allocation in multi-head beam-type SMD placement machines

This paper addresses a highly challenging scheduling problem faced in multi-head beam-type surface mounting devices (SMD) machines. An integrated mathematical model is formulated aiming to balance workloads over multiple heads as well as improving the traveling speed of the robotic arm by incorporating the appropriateness factors in the model to evaluate the compatibility of component-nozzle pairs. The proposed model is a bi-objective mixed integer nonlinear programming one, which is first converted into a linearized model and then directly solved by using the augmented epsilon constraint method for small problem instances. As the model is turned out to be NP-hard, we also develop a Multi-Objective Particle Swarm Optimization (MOPSO) algorithm to solve the model for medium and large-sized problem instances. The parameters of the proposed MOPSO are tuned by using the Taguchi Method and corresponding numerical results are provided.