Multi-objective Integer Linear Programming Research Articles

Sensitivity Analysis of the Cost Coefficients in Multiobjective Integer Linear Optimization

This paper considers sensitivity analysis of the cost coefficients in multiobjective integer linear programming problems. We define the sensitivity region as the set of simultaneous changes to the coefficients for which the efficient set and its structure remain the same. In particular, we require that the component-wise relation between efficient solutions is preserved and that inefficient solutions remain inefficient, and we show that this is sufficient for the efficient set to be the same upon changes. For a single coefficient, we show that a subset of the inefficient solutions can be excluded from consideration. More specifically, we prove that it suffices to inspect the inefficient solutions of a q-objective problem that are efficient in one of two related q + 1-objective problems. Finally, we show that the sensitivity region is a convex set (an interval). Our approach generalizes to simultaneous changes in multiple coefficients. For illustration, we consider mean-variance capital budgeting and determine the region for which the set of efficient portfolios remains the same, despite a misspecification or a change in the net present values of the projects. Further computational experience with multiobjective binary and integer knapsack problems demonstrates the general applicability of our technique. For instance, we obtain all sensitivity intervals for changes to single coefficients of biobjective problems with 500 binary variables in less than half an hour of CPU time by excluding a large number of inefficient solutions. In fact, the number of excluded solutions is above 100 orders of magnitude larger than the number of remaining solutions. This paper was accepted by Chung Piaw Teo, optimization. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.01406 .

A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging.

Due to the high pollution of the transportation sector, nowadays the role of electric vehicles has been noticed more and more by governments, organizations, and environmentally friendly people. On the other hand, the problem of electric vehicle routing (EVRP) has been widely studied in recent years. This paper deals with an extended version of EVRP, in which electric vehicles (EVs) deliver goods to customers. The limited battery capacity of EVs causes their operational domains to be less than those of gasoline vehicles. For this purpose, several charging stations are considered in this study for EVs. In addition, depending on the operational domain, a full charge may not be needed, which reduces the operation time. Therefore, partial recharging is also taken into account in the present research. This problem is formulated as a multi-objective integer linear programming model, whose objective functions include economic, environmental, and social aspects. Then, the preemptive fuzzy goal programming method (PFGP) is exploited as an exact method to solve small-sized problems. Also, two hybrid meta-heuristic algorithms inspired by nature, including MOSA, MOGWO, MOPSO, and NSGAII_TLBO, are utilized to solve large-sized problems. The results obtained from solving the numerous test problems demonstrate that the hybrid meta-heuristic algorithm can provide efficient solutions in terms of quality and non-dominated solutions in all test problems. In addition, the performance of the algorithms was compared in terms of four indexes: time, MID, MOCV, and HV. Moreover, statistical analysis is performed to investigate whether there is a significant difference between the performance of the algorithms. The results indicate that the MOSA algorithm performs better in terms of the time index. On the other hand, the NSGA-II-TLBO algorithm outperforms in terms of the MID, MOCV, and HV indexes.

A genetics algorithms for optimizing a function over the integer efficient set

In this paper, we propose an algorithm called Directional Exploration Genetic Algorithm (DEGA) to resolve a function Phi over the efficient set of a multi-objective integer linear programming problem (MOILP). DEGA algorithm belongs to evolutionary algorithms, which operate on the decision space by choosing the fastest improving directions that improve the objectives functions and Phi function. Two variants of this algorithm and a basic version of the genetic algorithm (BVGA) are performed and implemented in Python. Several benchmarks are carried out to evaluate the algorithm's performances and interesting results are obtained and discussed.

A new cutting plane method for lexicographic multi-objective integer linear programming

This work presents a new cutting plane method for lexicographic multi-objective integer linear programming (LMOILP). The method uses Grossone Methodology to reformulate a LMOILP problem into one having a single scalar objective function involving Grossone, as done in Cococcioni et al. (2018) (but in that case in the absence of the integer constraints). The problem, without the integer constraints, is solved using the GrossSimplex algorithm presented in Cococcioni et al. (2018) to find a candidate optimal solution. Here a novel cutting plane is introduced, named Gross-based Objective Function Cutting Plane whenever the optimal value of the Gross-scalar objective function is Gross-fractional. When it happens to be not Gross-fractional, cutting planes are generated using the Fractional Cutting Plane, derived from fractional components of the optimal solution. Moreover, by combining them, we proposed an algorithm that we called Gross-based Cutting Plane (GCP) method. It has been proved that it finds the optimal solution of a LMOILP problem and terminates after a finite number of iterations. To speed-up the GCP, at each iteration subsequent the first one, we re-use the optimal basis of the last continuous relaxation, by running the GrossDualSimplex algorithm. This is the well-known warm-start technique, which, however, needs specific attention due to the need to solve a linear problem having a right-hand side involving Grossone-based numbers. In the experimental part, we show the efficacy of the proposed approach.

Optimization of a quadratic programming problem over an integer efficient set

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms.

Access mechanism for period flows of non-deterministic end systems for time-sensitive networks

The IEEE 802.1Qbv standard schedules time-triggered (TT) flows (i.e., period flows) in a fixed TT Window each period. However,period flows generated by non-deterministic end systems exhibit significant jitter, whichleads to a mismatch between the generation times and the scheduled TT Windows. In the worst case scenario, this mismatch can cause an additional send delay of approximately one period in the flow. In this study,we model the send delay issue due to the maximum send delay requirement for all frames in a period flow not being satisfied simultaneously, and measure the jitter of period flows in a typical non-deterministic end system. Moreover, we propose an access mechanism for jittered period flows to schedule multiple conflict-free TT Windows in each period of flows at the source end system and control the send delay within the required delay tolerance. This mechanism enables deterministic access to jittered period flows, providing a prerequisite for reliable end-to-end transmission in the network. Moreover, this mechanism adopts a multi-objective integer linear programming (MILP) solver to optimize the TT Windows schedule. Furthermore, we establish the constraints and objective functions for the MILP solver and evaluate the mechanism with actual sampled frames in a real non-deterministic end system. Compared with the conventional fixed single TT Window and worst-case delay analysis mechanisms, the proposed mechanism satisfies the send delay requirements and considerably reduces the buffer usage at the source end system.

Identifying Causal Relationships in a Strategy Map Using ANP and Multi-Objective Integer Optimization Model

This paper presents a method for identifying causal relationships between strategic objectives within a strategy map of a Balanced Scorecard. Strategy maps are modeled as a network of strategic objectives (nodes) and causal relationships (directed arcs). The nodes are also grouped into clusters that represent the perspectives of a Balanced Scorecard: (a) Finances, Clients, Internal Processes and Growth and Learning. The method uses the Analytic Network Process (ANP) to establish the importance of every relationship and uses a multi-objective integer linear programming model to select the relationships to be included within a strategy map of a company. The method provides a method that optimizes the selection of the relationships to be included in a strategy map. An illustration of the application of the method in a manufacturing company is presented.

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Bicycle sharing station planning: From free-floating to geo-fencing

Bicycle sharing systems (BSSs) are expanding with unparalleled speed all over the world due to various advantages of cycling such as cost effectiveness, environmentally friendliness, capability to bridge other public transport modes and reduce private vehicle usage. One of the latest trend of BSS mode appears to be free-floating, which offers more flexibility in bike rental and return for cyclists in terms of service locations. However, despite the convenience brought by the free-floating BSS to the users, it also causes a headache for the authorities, largely attributed to its poor enforcement on regulating bad user behaviours, which significantly deteriorates the city scape. In order to preserve the benefits of BSSs and meanwhile solve the problems caused by free-floating BSS, this study develops a multi-objective integer non-linear programming (MOINLP) model to convert an existing free-floating BSS into a geo-fence-based one by determining the geo-fencing stations to be opened, their corresponding parking capacities and deployment of bikes. The three objectives of the developed MOINLP model are minimizing monetary cost, user dissatisfaction as well additional first-/last-mile walking distance of users. The MOINLP model is transformed into an equivalent multi-objective integer linear programming (MOILP) model. The augmented ε-constraint method (AUGMECON2) is customized to solve the MOILP model for exact Pareto optimal solutions while a tailored simulation-based heuristic method is proposed to solve large-scale problems. The numerical experiments have testified the good performance of our proposed solution method and we extend our analysis to Beijing BSS to gain planning insights and policy implications.

A Heuristically Assisted Deep Reinforcement Learning Approach for Network Slice Placement

Network Slice placement with the problem of allocation of resources from a virtualized substrate network is an optimization problem which can be formulated as a multi-objective Integer Linear Programming (ILP) problem. However, to cope with the complexity of such a continuous task and seeking for optimality and automation, the use of Machine Learning (ML) techniques appear as a promising approach. We introduce a hybrid placement solution based on Deep Reinforcement Learning (DRL) and a dedicated optimization heuristic based on the “Power of Two Choices” principle. The DRL algorithm uses the so-called Asynchronous Advantage Actor Critic (A3C) algorithm for fast learning, and Graph Convolutional Networks (GCN) to automate feature extraction from the physical substrate network. The proposed Heuristically-Assisted DRL (HA-DRL) allows for the acceleration of the learning process and substantial gain in resource usage when compared against other state-of-the-art approaches, as evidenced by evaluation results.

Multi-objective optimization for coverage aware energy consumption in wireless 3D video sensor network

This paper addresses the problem of minimizing energy consumption as well as maximizing the coverage area for 3D video sensor nodes deployed over a 2D target area in a wireless video sensor network. Reducing the number of active video sensor nodes helps decrease energy consumption, but decrease the coverage area as well as connectivity in the network. The problem is solved initially by a multi-objective integer linear programming based approach (APP_7) which provides an optimal solution but leads to intractability for a large-sized network due to the non-deterministic polynomial time-hard complexity of the problem. This fact motivates us to design a non-dominated sorting genetic algorithm-II based heuristic (APP_8), which generates a near-optimal solution and that is substantiated by comparing the simulation results of APP_7 with APP_8, and two existing greedy heuristics (EX_1, EX_11) for small-sized problem. Simulation experiments are conducted also to compare the results of APP_8 with EX_1 and EX_11 for large-sized network. Between two heuristics, EX_11/(EX_1) is better in terms of energy consumption/(area coverage). It has been observed that for the large-sized network (i.e. node density 100 on the target area of size 75 × 75 square meters) APP_8 is able to reduce energy consumption by 47.76% from EX_11, whereas area coverage is decreased by 0.51% from EX_1, which confirms the acceptability of APP_8 as the proper heuristic in such a scenario.

New [formula omitted]constraint methods for multi-objective integer linear programming: A Pareto front representation approach

Dealing with multi-objective problems by using generation methods has some interesting advantages since it provides the decision-maker with the complete information about the set of non-dominated criterion vectors (Pareto front) and a clear overview of the different trade-offs of the problem. However, providing many solutions to the decision-maker may also be overwhelming. As an alternative approach, showing a representative set of the Pareto front may be advantageous. Choosing such a representative set is by itself also a multi-objective problem that must consider the number of alternatives to present, the uniformity, and/or the coverage of the representation, to guarantee its quality. This paper proposes three algorithms for the representation problem for multi-objective integer linear programming problems with two or more objective functions, each one of them dealing with each dimension of the problem (cardinality, coverage, and uniformity). Such algorithms are all based on the ϵ-constraint approach. In addition, the paper also presents strategies to overcome poor estimations of the Pareto front bounds. The algorithms were tested on the ability to efficiently generate the whole Pareto front or a representation of it. The uniformity and cardinality algorithms proved to be very efficient both on binary and on integer problems, being amongst the best in the literature. Both coverage and uniformity algorithms provide good quality representations on their targeted objective, while the cardinality algorithm appears to be the most flexible, privileging uniformity for lower cardinality representations and coverage on higher cardinality.

Dynamic UPF placement and chaining reconfiguration in 5G networks

Network function virtualization (NFV) and multi-access edge computing (MEC) have become two crucial pillars in developing 5G and beyond networks. NFV promises cost-saving and fast revenue generation through dynamic instantiation and the scaling of virtual network functions (VNFs) according to time-varying service demands. Additionally, MEC provides considerable reductions in network response time and backhaul traffic since network functions and server applications can be deployed close to users. Nevertheless, the placement and chaining of VNFs at the network edge is challenging due to numerous aspects and attendant trade-offs. This paper addresses the problem of dynamic user plane function placement and chaining reconfiguration (UPCR) in a MEC environment to cope with user mobility while guaranteeing cost reductions and acceptable quality of service (QoS). The problem is formalized as a multi-objective integer linear programming model to minimize multiple cost components involved in the UPCR procedure. We propose a heuristic algorithm called dynamic priority and cautious UPCR (DPC-UPCR) to reduce the solution time complexity. Additionally, we devise a scheduler mechanism based on optimal stopping theory to determine the best reconfiguration time according to instantaneous values of latency violations and a pre-established QoS threshold. Our detailed simulation results evidence the efficiency of the proposed approaches. Specifically, the DPC-UPCR provides near-optimal solutions, within 15% of the optimum in the worst case, in significantly shorter times than the mathematical model. Moreover, the proposed scheduling method outperforms two scheduler baseline solutions regarding the number of reconfiguration events and QoS levels.

Optimization of the Cognitive Processes in a Virtual Classroom: A Multi-objective Integer Linear Programming Approach

A fundamental problem in the design of a classroom is to identify what characteristics it should have in order to optimize learning. This is a complex problem because learning is a construct related to several cognitive processes. The aim of this study is to maximize learning, represented by the processes of attention, memory, and preference, depending on six classroom parameters: height, width, color hue, color saturation, color temperature, and illuminance. Multi-objective integer linear programming with three objective functions and 56 binary variables was used to solve this optimization problem. Virtual reality tools were used to gather the data; novel software was used to create variations of virtual classrooms for a sample of 112 students. Using an interactive method, more than 4700 integer linear programming problems were optimally solved to obtain 13 efficient solutions to the multi-objective problem, which allowed the decision maker to analyze all the information and make a final choice. The results showed that achieving the best cognitive processing performance involves using different classroom configurations. The use of a multi-objective interactive approach is interesting because in human behavioral studies, it is important to consider the judgement of an expert in order to make decisions.

Warm-starting lower bound set computations for branch-and-bound algorithms for multi objective integer linear programs

In this paper we propose a generic branch-and-bound algorithm for solving multi-objective integer linear programming problems. In the recent literature, competitive frameworks has been proposed for bi-objective 0–1 problems, and many of these frameworks rely on the use of the linear relaxation to obtain lower bound sets. When increasing the number of objective functions, however, the polyhedral structure of the linear relaxation becomes more complex, and consequently requires more computational effort to obtain. In this paper we overcome this obstacle by speeding up the computations. To do so, in each branching node we use information available from its father node to warm-start a Bensons-like algorithm. We show that the proposed algorithm significantly reduces the CPU time of the framework on several different problem classes with three, four and five objective functions. Moreover, we point out difficulties that arise when non-binary integer variables are introduced in the models, and test our algorithm on problem that contains non-binary integer variables too.

Decision space robustness for multi-objective integer linear programming

In this article we introduce robustness measures in the context of multi-objective integer linear programming problems. The proposed measures are in line with the concept of decision robustness, which considers the uncertainty with respect to the implementation of a specific solution. An efficient solution is considered to be decision robust if many solutions in its neighborhood are efficient as well. This rather new area of research differs from robustness concepts dealing with imperfect knowledge of data parameters. Our approach implies a two-phase procedure, where in the first phase the set of all efficient solutions is computed, and in the second phase the neighborhood of each one of the solutions is determined. The indicators we propose are based on the knowledge of these neighborhoods. We discuss consistency properties for the indicators, present some numerical evaluations for specific problem classes and show potential fields of application.

Supplier management by distributing orders among new and existing suppliers: the methodology and its application to a fast fashion company

PurposeThis study aims to create a reliable, collaborative and sustainable business environment with suppliers of a company for providing high-quality and low-cost products on time. A supplier management system that sustains existing suppliers by sharing work based on systematic performance evaluation while developing the supplier base with potential suppliers is proposed.Design/methodology/approachBuilt on quantitative approaches, supplier management functions are integrated in the designed system. A quantitative strengths, weaknesses, opportunities and threats (SWOT) analysis is adapted for evaluating potential suppliers. A multi-objective integer linear programming (ILP) model is developed for the distribution of orders among selected potential and existing suppliers. A performance evaluation scheme based on an exponentially weighted moving average (EWMA) is proposed to evaluate and monitor suppliers' performance over time.FindingsProposed system develops a supplier base by methodically selecting and approving new suppliers, and a sustainable relationship with both new and existing suppliers is established based on performance over time. Decisions on retaining or removing suppliers from the base are objectively made by quantitative evaluations. Orders are fairly distributed among suppliers under the constraints imposed by the management. Dependence on a certain set of suppliers and its associated risks are reduced while agility in offering goods is enabled.Originality/valueBusiness processes for selecting new suppliers, distributing orders among all suppliers, evaluating and monitoring performance over time are quantitatively integrated to add value in operational decision-making. The proposed system is original in the holistic approach for managing and sustaining multiple suppliers of a company based on performance.

A robust multiobjective integrated master surgery schedule and surgical case assignment model at a publicly funded hospital

This paper considers an integrated operating room (OR) planning and advanced scheduling problem in a surgery department comprising several specialties that share a fixed number of ORs and post-surgery beds. It jointly considers the allocation of surgical specialties to OR blocks together with the assignment of the subsets of patients from each specialty’s waiting list to the OR blocks over a one-week planning horizon. By integrating the master surgical schedule problem (MSSP) and surgical case assignment problem (SCAP), we extend the stability of the scheduling process. We consider both uncertain surgery durations and emergency arrivals. The aim of this study is to maximize both patient service levels and hospital efficiency. A multi-objective integer linear programming (MOILP) model is developed to minimize patient waiting times as well as to maximize the OR utilization rates. An Lp-metric methodology is applied to solve the final integrated objective function. The MOILP is transformed into a robust optimization model using a transformation framework. In addition, we develop a two-stage stochastic integer linear programming model. The effectiveness of three models is compared and demonstrated through an extensive numerical experiment carried out on a large set of instances based on real data.

Exact Zoning Optimization Model for Marine Spatial Planning (MSP)

Marine spatial planning (MSP) has recently attracted more attention as an efficient decision support tool. MSP is a strategic and long-term process gathering multiple competing users of the ocean with the objective to simplify decisions regarding the sustainable use of marine resources. One of the challenges in MSP is to determine an optimal zone to locate a new activity while taking into account the locations of the other existing activities. Most approaches to spatial zoning are formulated as non-linear optimization models involving multiple objectives, which are usually solved using stochastic search algorithms, leading to sub-optimal solutions. In this paper, we propose to model the problem as a Multi-Objective Integer Linear Program. The model is developed for raster data and it aims at maximizing the interest of the area of the zone dedicated to the new activity while maximizing its spatial compactness. We study two resolution methods: first, a weighted-sum of the two objectives, and second, an interactive approach based on an improved augmented version of the ϵ-constraint method, AUGMECON2. To validate and study the model, we perform experiments on artificially generated data. Our experimental study shows that AUGMECON2 represents the most promising approach in terms of relevance and diversity of the solutions, compactness, and computation time.

Efficient solutions to the placement and chaining problem of User Plane Functions in 5G networks

This study attempts to solve the placement and chaining problem of 5G User Plane Functions (UPFs) in a Multi-access Edge Computing (MEC) ecosystem. The problem is formalized as a multi-objective Integer Linear Programming (ILP) model targeted at optimizing provisioning costs and quality of service. Our model takes into account several aspects of the system such as UPF-specific considerations, the Service Function Chain (SFC) requests topology (single and multiple branches), Virtual Network Function (VNF) order constraints, service demands, and physical network capacities. Since the formulated problem is NP-hard, two heuristic solutions are devised to enhance solution efficiency. Specifically, an algorithm called Priority and Cautious-UPF Placement and Chaining (PC-UPC) and a simulated annealing (SA) meta-heuristic are proposed. Through extensive simulation experiments, we evaluated the performance of the proposed solutions. The results revealed that our solutions outperformed the baselines (i.e., two greedy-based heuristics and a variant of the classical SA) and that we had obtained nearly optimal solutions with significant reductions in running time. Moreover, the PC-UPC algorithm can effectively avoid SFC rejections and improve provisioning costs by considering session requirements, current network conditions, and the effects of VNF mapping decisions. Additionally, the proposed SA approach incorporates several mechanisms (e.g., variable Markov chain length and restart–stop) that allow the improvement of not only the quality of the solutions but also their computation time.