Multiobjective Integer Programming Problems Research Articles

An efficient branch‐and‐bound algorithm to optimize a function over a nondominated set

AbstractThis study introduces an algorithm based on the branch‐and‐bound approach for optimizing a main function over the nondominated set of a multiobjective integer programming (MOIP) problem. Initially, is optimized within the feasible solution set of the MOIP. A new efficiency test combining Benson's test with is then developed using an auxiliary optimization program. This program provides both an efficient solution and a lower bound for . Moreover, this solution is the best one for when compared to its alternative solutions for MOIP. Subsequently, efficient cuts are incorporated into the criteria space to eliminate dominated points. Furthermore, the algorithm is tailored to handle scenarios where the objective involves optimizing a linear combination of multiobjective programming criteria over the nondominated set. The study concludes by showcasing the superior performance of the proposed two algorithms through comparison with existing approaches on well‐known problem instances from the literature.

A simple, efficient and versatile objective space algorithm for multiobjective integer programming

In the last years a multitude of algorithms have been proposed to solve multiobjective integer programming problems. However, only few authors offer open-source implementations. On the other hand, new methods are typically compared to code that is publicly available, even if this code is known to be outperformed. In this paper, we aim to overcome this problem by proposing a new state-of-the-art algorithm with an open-source implementation in C++. The underlying method falls into the class of objective space methods, i.e., it decomposes the overall problem into a series of scalarized subproblems that can be solved with efficient single-objective IP-solvers. It keeps the number of required subproblems small by avoiding redundancies, and it can be combined with different scalarizations that all lead to comparably simple subproblems. Our algorithm bases on previous results but combines them in a new way. Numerical experiments with up to ten objectives validate that the method is efficient and that it scales well to higher dimensional problems.

A multi-objective joint optimisation method for simultaneous part family formation and configuration design in delayed reconfigurable manufacturing system (D-RMS)

ABSTRACT In the era of Industry 4.0, the demand fluctuation has become fiercer due to the characteristics of diversification, customisation, and uncertainty. Reconfigurability of manufacturing systems has been proven to be a useful and necessary feature when it comes to handling demand uncertainty. This feature can be achieved through the implementation of reconfigurable manufacturing system (RMS) and delayed reconfigurable manufacturing system (D-RMS). D-RMS is a subclass of RMS that focuses primarily on improving the convertibility of the manufacturing system. The two main phases involved in implementing D-RMS are part family formation and configuration design. Therefore, we proposed a multi-objective joint optimisation method of part family formation and configuration design according to the philosophy of D-RMS. Firstly, we develop a multi-objective joint optimisation model that takes into account investment cost, reconfiguration cost, similarity coefficient, and delayed reconfiguration to optimise the part family and configuration of D-RMS simultaneously. Three types of machine tools namely dedicated machine tools, flexible machine tools, and reconfigurable machine tools are considered in the optimisation model. Secondly, the non-dominated sorting genetic algorithm-III (NSGA-III) is adopted to solve the proposed multi-objective integer programming problem. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed multi-objective joint optimisation method.

Parallel and Collaborative Passenger Flow Control of Urban Rail Transit Under Comprehensive Emergency Situation

The artificial cooperative control methods are used to fully utilize the train capacity and reduce the passenger delays by coordinating the number of entering and boarding passengers at each station, but it ignores the potential risk associated with the aggregation attributes of passengers in the station under emergency and the unfairness caused by the imbalance of travel opportunities for different origin–destination (OD) passengers. A new parallel and cooperative control model under comprehensive emergency situation is constructed from the perspective of guaranteeing travel safety, improving travel fairness, and increasing travel efficiency to generate passenger flow control strategies. The model aims to maximize passenger turnovers, minimize passenger delays and reduce the difference in the ratio of stranded passengers at stations. The complex multi-objective integer programming problem involving nonlinear equation constraints is difficult to solve numerically. A dynamic programming model taking train sequence as stage variable is then constructed as well as an NSGA-II algorithm with multiple improved strategies (MIS-NSGA-II) is proposed to solve the model. Numerical experiments on a line of Beijing Rail Transit show that the MIS-NSGA-II outperforms the traditional NSGA-II, and that increasing the overload coefficient of the train and the tolerance coefficient of each area in the station can further alleviate passenger congestion and promote the operation efficiency. Compared with other control strategies mentioned in this study, the proposed control model from a comprehensively perspective, which can ensure the travel safety and promote the travel efficiency while forcefully decongesting the bottleneck areas and improving the fairness of passenger travel.

Novo rešenje problema višekriterijumskog celobrojnog programiranja pomoću višekriterijumske optimizacije zasnovane na verovatnoći

Introduction/purpose: In this paper, a new solution for solving a multiobjective integer programming problem with probabilistic multi - objective optimization is formulated. Furthermore, discretization by means of the good lattice point and sequential optimization are employed for a successive simplifying treatment and deep optimization. Methods: In probabilistic multi - objective optimization, a new concept of preferable probability has been introduced to describe the preference degree of each performance utility of a candidate; each performance utility of a candidate contributes a partial preferable probability and the product of all partial preferable probabilities deduces the total preferable probability of a candidate; the total preferable probability thus transfers a multi-objective problem into a single-objective one. Discretization by means of the good lattice point is employed to conduct discrete sampling for a continuous objective function and sequential optimization is used to perform deep optimization. At first, the requirements of integers in the treatment could be given up so as to simply conduct above procedures. Finally, the optimal solutions of the input variables must be rounded to the nearest integers. Results: This new scheme is used to deal with two production problems, i.e., maximizing profit while minimizing pollution and determining a purchasing plan for spending as little money as possible while getting as large amount of raw materials as possible. Promising results are obtained for the above two problems from the viewpoint of the probability theory for simultaneous optimization of multiple objectives. Conclusion: This method properly considers simultaneous optimization of multiple objectives in multi-objective integer programming, which naturally reflects the essence of multi-objective programming, and opens a new way of solving multi-objective problems.

Exact Methods for Multi-Objective Integer Nonlinear Programming

Multi-objective integer nonlinear programming (MOINLP) problems are multi-objective integer programming problems with at least one nonlinear objective function or constraint. To date, MOINLP problem has not been exactly solved. Although traditional ε-constraint method can be used to solve MOINLP problem, the obtained solution may not be Pareto-optimal. To overcome this shortcoming, a basic ε-constraint method (BEM) is proposed to solve MOINLP problem exactly. However, the time complexity of BEM is as high as O ( ( p − 1 ) M 2 N ) , where p, M, and N are the numbers of objectives, Pareto-optimal solutions, and feasible solutions, respectively. For this reason, an improved BEM (IBEM) is developed whose time complexity is O ( M 2 N ) . That is, the time complexity of IBEM for solving MOINLP problem is equal to solving the single-objective one. Finally, to avoid using all the feasible solutions (N) in obtaining each Pareto-optimal solution, three methods to eliminate dominated solutions effectively are used before performing IBEM. The test results illustrate that our method can not only solve MOINLP problem exactly but also has high efficiency.

Joint access point selection and resource allocation in MEC-assisted network: A reinforcement learning based approach

A distributed reinforcement learning (RL) based resource management framework is proposed for a mobile edge computing (MEC) system with both latency-sensitive and latency-insensitive services. We investigate joint optimization of both computing and radio resources to achieve efficient on-demand matches of multi-dimensional resources and diverse requirements of users. A multi-objective integer programming problem is formulated by two subproblems, i.e., access point (AP) selection and subcarrier allocation, which can be solved jointly by our proposed distributed RL-based approach with a heuristic iteration algorithm. The proposed algorithm allows for the reduction in complexity since each user needs to consider only its own selection of AP without knowing full global information. Simulation results show that our algorithm can achieve near-optimal performance while reducing computational complexity significantly. Compared with other algorithms that only optimize either of the two sub-problems, the proposed algorithm can serve more users with much less power consumption and content delivery latency.

Split algorithms for multiobjective integer programming problems

We consider split algorithms that partition the objective function space into p or p−1 dimensional regions so as to search for nondominated points of multiobjective integer programming problems, where p is the number of objectives. We provide a unified approach that allows different split strategies to be used within the same algorithmic framework with minimum change. We also suggest an effective way of making use of the information on subregions when setting the parameters of the scalarization problems used in the p-split structure. We compare the performances of variants of these algorithms both as exact algorithms and as solution approaches under time restriction, considering the fact that finding the whole set may be computationally infeasible or undesirable in practice. We demonstrate through computational experiments that while the (p−1)-split structure is superior in terms of overall computational time, the p-split structure provides significant advantage under time/cardinality limited settings in terms of representativeness, especially with adaptive parameter setting and/or a suitably chosen order for regions to be explored.

Algorithms for generating Pareto fronts of multi-objective integer and mixed-integer programming problems

Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present article shows that certain algorithms that were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are tested by means of two-, three- and four-objective integer and mixed-integer problems, and comparisons are made. In particular, a new four-objective algorithm is used to solve a rocket injector design problem with a discrete variable, which is a challenging mixed-integer programming problem.

Solving the multi-objective stochastic interval-valued linear fractional integer programming problem

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

Multiobjective Optimization-Aided Decision-Making System for Large-Scale Manufacturing Planning.

This work is geared toward a real-world manufacturing planning (MP) task, whose two objectives are to maximize the order fulfillment rate and minimize the total cost. More important, the requirements and constraints in real manufacturing make the MP task very challenging in several aspects. For example, the MP needs to cover many production components of multiple plants over a 30-day horizon, which means that it involves a large number of decision variables. Furthermore, the MP task's two objectives have extremely different magnitudes, and some constraints are difficult to handle. Facing these uncompromising practical requirements, we introduce an interactive multiobjective optimization-based MP system in this article. It can help the decision maker reach a satisfactory tradeoff between the two objectives without consuming massive calculations. In the MP system, the submitted MP task is modeled as a multiobjective integer programming (MOIP) problem. Then, the MOIP problem is addressed via a two-stage multiobjective optimization algorithm (TSMOA). To alleviate the heavy calculation burden, TSMOA transforms the optimization of the MOIP problem into the optimization of a series of single-objective problems (SOPs). Meanwhile, a new SOP solving strategy is used in the MP system to further reduce the computational cost. It utilizes two sequential easier SOPs as the approximator of the original complex SOP for optimization. As part of the MP system, TSMOA and the SOP solving strategy are demonstrated to be efficient in real-world MP applications. In addition, the effectiveness of TSMOA is also validated on benchmark problems. The results indicate that TSMOA as well as the MP system are promising.

Solution approaches for equitable multiobjective integer programming problems

We consider multi-objective optimization problems where the decision maker (DM) has equity concerns. We assume that the preference model of the DM satisfies properties related to inequity-aversion, hence we focus on finding nondominated solutions in line with the properties of inequity-averse preferences, namely the equitably nondominated solutions. We discuss two algorithms for finding good subsets of equitably nondominated solutions. The first approach is an extension of an interactive approach developed for finding the most preferred nondominated solution when the utility function is assumed to be quasiconcave. We find the most preferred equitably nondominated solution when the utility function is assumed to be symmetric quasiconcave. In the second approach we generate an evenly distributed subset of the set of equitably nondominated solutions to be considered further by the DM. We show the computational feasibility of the two algorithms on equitable multi-objective knapsack problem, in which projects in different categories are to be funded subject to a limited budget. We perform experiments to show and discuss the performances of the algorithms.

Enhancement of the Improved Recursive Method for Multi-objective Integer Programming Problem

In this paper, we developed a new algorithm to find the set of a non-dominated points for a multi-objective integer programming problem. The algorithm is an enhancement on the improved recursive method where the authors have used a lexicographic method for analysis. In this approach a sum of two objectives is considered as one weighted sum objective for each iteration. Computational results show that the proposed approach outperforms the currently available results obtained by the improved recursive method with respect to CPU time and the number of integer problems solved to identify all non-dominated points. Many problems such as assignment, knapsack and travelling salesman have been investigated on different sized problems. The benefit of this approach becomes more visible with the increase in the number of objective functions.

Biobjective optimization over the efficient set of multiobjective integer programming problem

<p style='text-indent:20px;'>In this article, an exact method is proposed to optimize two preference functions over the efficient set of a multiobjective integer linear program (MOILP). This kind of problems arises whenever two associated decision-makers have to optimize their respective preference functions over many efficient solutions. For this purpose, we develop a branch-and-cut algorithm based on linear programming, for finding efficient solutions in terms of both preference functions and MOILP problem, without explicitly enumerating all efficient solutions of MOILP problem. The branch and bound process, strengthened by efficient cuts and tests, allows us to prune a large number of nodes in the tree to avoid many solutions. An illustrative example and an experimental study are reported.

Multiobjective Integer Programming: Synergistic Parallel Approaches

Exactly solving multiobjective integer programming (MOIP) problems is often a very time-consuming process, especially for large and complex problems. Parallel computing has the potential to signifi...

An exact parallel objective space decomposition algorithm for solving multi-objective integer programming problems

The set of all nondominated solutions for a multi-objective integer programming (MOIP) problem is finite if the feasible region is bounded, and it may contain unsupported solutions. Finding these sets is NP-hard for most MOIP problems and current methods are unable to scale with the number of objectives. We propose a deterministic exact parallel algorithm for solving MOIP problems with any number of objectives. The proposed algorithm generates the full set of nondominated solutions based on intelligent iterative decomposition of the objective space utilizing a particular scalarization scheme. The algorithm relies on a set of rules that exploits regional dominance relations among the decomposed partitions for pruning. These expediting rules are both used as part of a pre-solve step as well as judiciously employed throughout the parallel running threads. Using an extensive test-bed of MOIP instances with three, four, five, and six objectives, the performance of the proposed algorithm is evaluated and compared with leading benchmark algorithms for MOIPs. Results of the experimental study demonstrate the effectiveness of the proposed algorithm and the computational advantage of its parallelism.

Pricing and Collaboration in Last Mile Delivery Services

Recently, last mile delivery has emerged as an essential process that greatly affects the opportunity of obtaining delivery service market share due to the rapid increase in the business-to-consumer (B2C) service market. Express delivery companies are investing to expand the capacity of hub terminals to handle increasing delivery volume. As for securing massive delivery quantity by investment, companies must examine the profitability between increasing delivery quantity and price. This study proposes two strategies for a company’s decision making regarding the adjustment of market density and price by developing a pricing and collaboration model based on the delivery time of the last mile process. A last mile delivery time function of market density is first derived from genetic algorithm (GA)-based simulation results of traveling salesman problem regarding the market density. The pricing model develops a procedure to determine the optimal price, maximizing the profit based on last mile delivery time function. In addition, a collaboration model, where a multi-objective integer programming problem is developed, is proposed to sustain long-term survival for small and medium-sized companies. In this paper, sensitivity analysis demonstrates the effect of delivery environment on the optimal price and profit. Also, a numerical example presents four different scenarios of the collaboration model to determine the applicability and efficiency of the model. These two proposed models present managerial insights for express delivery companies.

A Lagrangian Relaxation for a Fuzzy Random EPQ Problem with Shortages and Redundancy Allocation: Two Tuned Meta-heuristics

This paper develops an economic production quantity model for a multi-product multi-objective inventory control problem with fuzzy-stochastic demand and backorders. In this model, the annual demand is represented by trapezoidal fuzzy random numbers. The centroid defuzzification and the expected value methods are applied to defuzzify and make decisions in a random environment. In the case where the warehouse space is limited, the Lagrangian relaxation procedure is first employed to determine the optimal order and the maximum backorder quantities of the products such that the total inventory cost is minimized. The optimal solution obtained by the proposed approach is compared with that obtained by the traditional deterministic method. Moreover, a sensitivity analysis presents the rationality of the solution. Then, the model is extended to a multi-objective integer programming problem in which the optimal numbers of redundant production machines are determined to maximize the production system reliability. In the second proposed model, several constraints are considered to fit real-world situations. As the second model is developed for an NP-hard problem and hence cannot be solved using exact methods in a reasonable computational time, a multi-objective evolutionary algorithm called non-dominated sorting genetic algorithm-II (NSGA-II) is employed to provide Pareto front solutions. Due to non-availability of benchmark in the literature, another multi-objective evolutionary algorithm called non-dominated ranking genetic algorithm is implemented as well to validate the obtained results and evaluate the performance of NSGA-II. In addition, the Taguchi method is used to calibrate the parameters of both algorithms for better performance.

On nadir points of multiobjective integer programming problems

In this study, we consider the nadir points of multiobjective integer programming problems. We introduce new properties that restrict the possible locations of the nondominated points necessary for computing the nadir points. Based on these properties, we reduce the search space and propose an exact algorithm for finding the nadir point of multiobjective integer programming problems. We present an illustrative example on a three objective knapsack problem. We conduct computational experiments and compare the performances of two recent algorithms and the proposed algorithm.

Multiobjective Location Model Design Based on Government Subsidy in the Recycling of CDW

With the generation of a large amount of construction and demolition waste (CDW), many scholars have recently paid more attention to the recycling of CDW. In this paper, we design a classification recycling method based on the degree of CDW availability in the recycling of CDW. Considering the important role of the government in reverse logistics, a model of reverse logistics network based on the trade-off between cost and recycling rate is put forward, which is subject to government subsidy. The model includes the location of classification processing center and ensures the best route of transportation. Then, the improved particle swarm optimization algorithm is applied to solve the model to get Pareto frontier by transforming it into a multiobjective integer-programming problem. As a case study, the results of the statistical modeling used in this study indicate the feasibility of the model. Finally, according to the sensitivity analysis of government’s subsidy, we evaluate the effectiveness of this program and its applicability.