Multi-objective Integer Programming Research Articles

An Interactive Fuzzy Satisficing Method for Random Fuzzy Multiobjective Integer Programming Problems through Probability Maximization with Possibility

This paper considers multiobjective integer programming problems where each coefficient of the objective functions is expressed by a random fuzzy variable. A new decision making model is proposed by incorporating the concept of probability maximization into a possibilistic programming model. For solving transformed deterministic problems, genetic algorithms with double strings for nonlinear integer programming problems are introduced. An interactive fuzzy satisficing method is presented for deriving a satisficing solution to a decision maker by updating the reference probability levels. An illustrative numerical example is provided to clarify the proposed method.

Multi-Objective Integer Programming: An Improved Recursive Algorithm

This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of Özlen and Azizoğlu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.

$\varepsilon$-Constraint and Fuzzy Logic-Based Optimization of Hazardous Material Transportation via Lane Reservation

With economic development, a great amount of hazardous material is shipped in the transport network every day. Hazardous material transportation is well known for its high potential risk. An accident can cause very serious economic damage and will have a negative impact on public health and the environment over the long term. Transporting hazardous materials on special lanes can reduce the risk. However, a lane reservation strategy may worsen traffic conditions for other vehicles. This paper investigates a hazardous material transportation problem with lane reservation. The problem lies in how to choose lanes to be reserved in the network and select the path for each hazardous material shipment from the reserved lanes. The goal is to obtain the best compromise between the impact on normal traffic and the transportation risk. A multiobjective integer programming model is presented for the new problem. Then, an algorithm is developed based on the ε -constraint method and a fuzzy-logic-based approach. Pareto optimal solutions are obtained by the former, and a preferred solution is selected by the fuzzy-logic-based approach. Computational results demonstrate the efficiency of the proposed algorithm using an instance based on a real network topology and randomly generated instances.

An improved version of the augmented ε-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems

Generation (or a posteriori) methods in Multi-Objective Mathematical Programming (MOMP) is the most computationally demanding category among the MOMP approaches. Due to the dramatic increase in computational speed and the improvement of Mathematical Programming algorithms the generation methods become all the more attractive among today’s decision makers. In the current paper we present the generation method AUGMECON2 which is an improvement of our development, AUGMECON. Although AUGMECON2 is a general purpose method, we will demonstrate that AUGMECON2 is especially suitable for Multi-Objective Integer Programming (MOIP) problems. Specifically, AUGMECON2 is capable of producing the exact Pareto set in MOIP problems by appropriately tuning its running parameters. In this context, we compare the previous and the new version in a series of new and old benchmarks found in the literature. We also compare AUGMECON2’s performance in the generation of the exact Pareto sets with established methods and algorithms based on specific MOIP problems (knapsack, set packing) and on published results. Except from other Mathematical Programming methods, AUGMECON2 is found to be competitive also with Multi-Objective Meta-Heuristics (MOMH) in producing adequate approximations of the Pareto set in Multi-Objective Combinatorial Optimization (MOCO) problems.

New Reference-Neighbourhood Scalarization Problem for Multiobjective Integer Programming

Abstract Scalarization is a frequently used approach for finding efficient solutions that satisfy the preferences of the Decision Maker (DM) in multicriteria optimization. The applicability of a scalarization problem to solve integer multicriteria problems depends on the possibilities it provides for the decrease of the computing complexity in finding optimal solutions of this class of problems. This paper presents a reference-neighbourhood scalarizing problem, possessing properties that make it particularly suitable for solving integer problems. One of the aims set in this development has also been the faster obtaining of desired criteria values, defined by the DM, requiring no additional information by him/her. An illustrative example demonstrates the features of this scalarizing problem.

A multi-objective optimisation model for university course timetabling problem using a mixed integer dynamic non-linear programming

This paper proposes a multi-objective integer programming formulation to solve course timetabling problem at universities. The most important objective in the proposed model is the collegians interferences. In other words, the student utility to take a course and the professor utility to present a course are combined simultaneously in objective function. Based on this definition, several time constraints are added. In this model, it is assumed that there are only courses with two and three units. At first it is supposed that each course can be presented in one group and thus the problem is solved by GAMS. Then, the number of course groups is devolved to the software by fitting the objective function value on a predetermined one. Computational results show that the best value for each courses group will be one group.

Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost

In stratified sampling, usually the cost function is taken as a linear function of sample sizes n h . In many practical situations, the linear cost function does not approximate the actual cost incurred adequately. For example, when the cost of travelling between the units selected in the sample within a stratum is significant, instead of a linear cost function, a cost function that is quadratic in √n h will be a more close approximation to the actual cost. In this paper, the problem is formulated as multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population. A numerical example is also presented to illustrate the computational details.

Multi-Objective Optimization of Human Resource Allocation in Production Cells with Different Training Modes

Aiming at the difference of the people as a particularity resource。In this paper ,the personnel training mode is divided into junior and senior, and a multi-objective integer programming model is established at the lowest cost of staff training, the highest man-machine adaptability degree and minimum personnel workload. Calculating example of a real production cell is presented. The results show that the model is correct and the necessity for classification of training modes.The model can help the management to adopt reasonable training mode and achieve desirable objectives.

Optimizing a multi-echelon supply chain network flow using nonlinear fuzzy multi-objective integer programming: Genetic algorithm approach

Article history: Received March 19, 2012 Received in Revised form June, 20, 2012 Accepted 22 June 2012 Available online June 24 2012 The aim of this paper is to present mathematical models optimizing all materials flows in supply chain. In this research a fuzzy multi-objective nonlinear mixedinteger programming model with piecewise linear membership function is applied to design a multi echelon supply chain network (SCN) by considering total transportation costs and capacities of all echelons with fuzzy objectives. The model that is proposed in this study has 4 fuzzy functions. The first function is minimizing the total transportation costs between all echelons (suppliers, factories, distribution centers (DCs) and customers). The second one is minimizing holding and ordering cost on DCs. The third objective is minimizing the unnecessary and unused capacity of factories and DCs via decreasing variance of transported amounts between echelons. The forth is minimizing the number of total vehicles that ship the materials and products along with SCN. For solving such a problem, as nodes increases in SCN, the traditional method does not have ability to solve large scale problem. So, we applied a Meta heuristic method called Genetic Algorithm. The numerical example is real world applied and compared the results with each other demonstrate the feasibility of applying the proposed model to given problem, and also its advantages are discussed. © 2012 Growing Science Ltd. All rights reserved.

Finding all nondominated points of multi-objective integer programs

We develop exact algorithms for multi-objective integer programming (MIP) problems. The algorithms iteratively generate nondominated points and exclude the regions that are dominated by the previously-generated nondominated points. One algorithm generates new points by solving models with additional binary variables and constraints. The other algorithm employs a search procedure and solves a number of models to find the next point avoiding any additional binary variables. Both algorithms guarantee to find all nondominated points for any MIP problem. We test the performance of the algorithms on randomly-generated instances of the multi-objective knapsack, multi-objective shortest path and multi-objective spanning tree problems. The computational results show that the algorithms work well.

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.

The DNA Genetic Algorithm Applied for Solving Optimal Placement of Sensors

In order to solve optimal placement of bridge sensors based on the modal information, a multi-objective integer programming expected value model is established and the number of modal is considered as a stochastic variable in this paper, and here Fisher and MAC matrix are combined as the objective functions . Since DNA Genetic algorithm has the merits of plentiful coding, and decoding, conveying complex knowledge flexibly, these merits and the technique of stochastic simulation are also combined, which for estimating stochastic integer programming expected value models problem. And finally the feasibility of the algorithm is showed by XuGe Bridge as an example.

Stochastic optimization models for ground delay program planning with equity–efficiency tradeoffs

Bad weather is a major contributor to air traffic delays. The ration-by-distance algorithm was proposed to increase the efficiency of ground delay program (GDP) planning in situations where there is uncertainty regarding the duration of the weather conditions that motivated the GDP. This algorithm, however, can treat shorter flights unfairly. We describe a two-stage stochastic, multi-objective integer program for GDP planning. This multi-objective approach allows one to balance equity and efficiency. We provide experimental evidence showing the effectiveness of the solutions generated.

Network Partition of Switched Industrial Ethernet by Using Novel Particle Swarm Optimization

In this paper, a novel particle swarm optimization is proposed to solve integer programming problem, which is name as an adaptive inertia weight integer programming particle swarm optimization (AIWIP-PSO). Then it is applied to solve the industrial Ethernet network partition problem, after the problem is formulated as multi-objective integer programming problem. This multi-objective integer programming problem includes two objectives: one is the minimum of the inter-network communication; the other is the evenness of the network traffic between the sub-networks. Furthermore, the switch capability must be considered when assigning devices to sub-networks. Two network partition cases are chosen for simulations. The results obtained by using AIWIP-PSO are compared with those from genetic algorithm. Simulation results show that the AIWIP-PSO is very effective and outperforms genetic algorithm in terms of the performance index

An Interactive Fuzzy Satisficing Method for Multiobjective Stochastic Integer Programming with Simple Recourse

This paper considers multiobjective integer programming problems involving random variables in constraints. Using the concept of simple recourse, the formulated multiobjective stochastic simple recourse problems are transformed into deterministic ones. For solving transformed deterministic problems efficiently, we also introduce genetic algorithms with double strings for nonlinear integer programming problems. Taking into account vagueness of judgments of the decision maker, an interactive fuzzy satisficing method is presented. In the proposed interactive method, after determineing the fuzzy goals of the decision maker, a satisficing solution for the decision maker is derived efficiently by updating the reference membership levels of the decision maker. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Daily scheduling of nurses in operating suites

This article provides a new multi-objective integer programming model for the daily scheduling of nurses in operating suites. The model is designed to assign nurses to different surgery cases based on their specialties and competency levels, subject to a series of hard and soft constraints related to nurse satisfaction, idle time, overtime, and job changes during a shift. To find solutions, two methodologies were developed. The first is based on the idea of a solution pool and the second is a variant of modified goal programming. The model and the solution procedures were validated using real data provided by the University of Texas MD Anderson Cancer Center in Houston, Texas. The results show that the two methodologies can produce schedules that satisfy all demand with 50% less overtime and 50% less idle time when benchmarked against current practice.

Risk approaches for delivering disaster relief supplies

We consider the problem of designing the logistic system to assure adequate distribution of relief aid in a post-natural-disaster situation, when damages to infrastructure may disrupt the delivery of relief aid. The problem is formulated as a multi-objective optimization problem, encompassing three objective functions of central interest in such problems. The first objective function is a measure of risk (various forms of such risk are analyzed). The second objective function measures the coverage provided by the logistic system in the distribution of relief aid to disaster victims. The third objective function represents total travel time. We focus on the risk of delivery tours for relief supplies, where risk here captures the threat that potential tours become impassable after the natural hazard event. In order to cope with a range of different natural disasters and policy objectives, we develop five approaches emphasizing different measures of tour-dependent risk. To cover both earthquake and flood risks, we consider correlated as well as uncorrelated risk measures. We develop a two-phase solution approach to reflect the dictates of real-world disaster relief motivating this analysis. The first phase generates potentially Pareto-optimal solutions to the overall multi-objective logistic design problem with respect to three objectives. For any given risk measure, the first-phase design problem is formulated as a multi-objective integer program and a memetic algorithm is proposed as the solution approach. The second phase is an enrichment procedure to generate a broader range of potentially Pareto-optimal alternatives. The suggested approach is tested on real-world data from the province of Manabi in Ecuador and the results associated with the different risk measures are analyzed to illustrate the value of the proposed approach for the design of disaster relief networks.

A logistics model for emergency supply of critical items in the aftermath of a disaster

This paper proposes a logistics model for delivery of prioritized items in disaster relief operations. It considers multi-items, multi-vehicles, multi-periods, soft time windows, and a split delivery strategy scenario, and is formulated as a multi-objective integer programming model. To effectively solve this model we limit the number of available tours. Two heuristic approaches are introduced for this purpose. The first approach is based on a genetic algorithm, while the second approach is developed by decomposing the original problem. We compare these two approaches via a computational study. The multi-objective problem is converted to a single-objective problem by the weighted sum method. A case study is presented to illustrate the potential applicability of our model. Also, presented is a comparison of our model with that proposed in a recent paper by Balcik et al. [6]. The results show that our proposed model outperforms theirs in terms of delivering prioritized items over several time periods.

An optimization approach to gene stacking

We present a multi-objective integer programming model for the gene stacking problem, which is to bring desirable alleles found in multiple inbred lines to a single target genotype. Pareto optimal solutions from the model provide strategic stacking schemes to maximize the likelihood of successfully creating the target genotypes and to minimize the number of generations associated with a stacking strategy. A consideration of genetic diversity is also incorporated in the models to preserve all desirable allelic variants in the target population. Although the gene stacking problem is proved to be NP-hard, we have been able to obtain Pareto frontiers for smaller sized instances within one minute using the state-of-the-art commercial computer solvers in our computational experiments.

A new complexity result on multiobjective linear integer programming using short rational generating functions

This paper presents a new complexity result for solving multiobjective integer programming problems. We prove that encoding the entire set of nondominated solutions of the problem in a short sum of rational functions is polynomially doable, when the dimension of the decision space is fixed. This result extends a previous result presented in De Loera et al. (INFORMS J. Comput. 21(1):39–48, 2009) in that there the number of the objective functions is assumed to be fixed whereas ours allows this number to vary.