Assigning cooperating UAVs to simultaneous tasks on consecutive targets using genetic algorithms

A problem of assigning multiple agents to simultaneously perform cooperative tasks on consecutive targets is posed as a new combinatorial optimization problem. The investigated scenario consists of multiple ground moving targets prosecuted by a team of unmanned aerial vehicles (UAVs). The team of agents is heterogeneous, with each UAV carrying designated sensors and all but one carry weapons as well. To successfully prosecute each target it needs to be simultaneously tracked by two UAVs and attacked by a third UAV carrying a weapon. Only for small-sized scenarios involving not more than a few vehicles and targets the problem can be solved in sufficient time using classical combinatorial optimization methods. For larger-sized scenarios the problem cannot be solved in sufficient time using these methods due to timing constraints on the simultaneous tasks and the coupling between task assignment and path planning for each UAV. A genetic algorithm (GA) is proposed for efficiently searching the space of feasible solutions. A matrix representation of the chromosomes simplifies the encoding process and the application of the genetic operators. To further simplify the encoding, the chromosome is composed of sets of multiple genes, each corresponding to the entire set of simultaneous assignments on each target. Simulation results show the viability of the proposed assignment algorithm for different sized scenarios. The sensitivity of the performance to variations in the GA tuning parameters is also investigated.     

Heuristics for multimode scheduling problems with dedicated resources

We study scheduling problems with multiple modes and dedicated resources arising in production and project management, which constitute a special class of the general multimode resource-constrained project scheduling problem. A task may require simultaneously a set of discrete, renewable resources to be processed and the processing can be performed in different modes, that is with different resource sets, processing times, or costs. Precedence constraints can exist among tasks. The total budget that can be allocated to the project can be limited. The problem consists of identifying a mode for each task and a starting time for its processing respecting precedence, resource, and budget constraints. A graph model and an iterative solution scheme are presented. Specific heuristic algorithms for the cases with and without budget constraints are given and computational results are discussed.     

An efficient solution approach for real-world driver scheduling problems in urban bus transportation

The driver scheduling problem in public transportation is defined in the following way. There is a set of operational tasks, and the goal is to define the sequence of these tasks as shifts in such a way that every task must be assigned to a shift without overlapping. In real-world situations several additional constraints need to be considered, which makes large practical problems challenging to be solved efficiently. In practice it is also an important request with respect to a public transportation scheduling system to offer several versions of quasi-optimal solutions. In this paper we present an efficient driver scheduling solution methodology which is flexible in the above sense.     

A global MILP model for FMS scheduling

Flexible manufacturing systems (FMS) require intelligent scheduling strategies to achieve their principal benefit — combining high flexibility with high productivity. A mixed-integer linear programming model (MILP) is presented here for FMS scheduling. The model takes a global view of the problem and specifically takes into account constraints on storage and transportation. Both of these constrained resources are critical for practical FMS scheduling problems and are difficult to model. The MILP model is explained and justified and its complexity is discussed. Two heuristic procedures are developed, based on an analysis of the global MILP model. Computational results are presented comparing the performance of the different solution strategies. The development of iterative global heuristics based on mathematical programming formulations is advocated for a wide class of FMS scheduling problems.     

Optimal periodic scheduling of multipurpose batch plants

A rigorous mathematical programming framework for the scheduling of multipurpose batch plants operated in a cyclic mode is presented. The proposed formulation can deal with batch operations described by complex processing networks, involving shared intermediates, material recycles, and multiple processing routes to the same end-product or intermediate. Batch aggregation and splitting are also allowed. The formulation permits considerable flexibility in the utilisation of processing equipment and storage capacity, and accommodates problems with limited availability of utilities. The scheduling problem is formulated as a large mixed integer linear program (MILP). For a given cycle time, it is shown that it is sufficient for the formulation to characterize a single cycle of the periodic schedule despite the existence of tasks that span two successive cycles. The optimal cycle time is determined by solving a sequence of fixed cycle time problems. The MILP is solved by a branch-and-bound algorithm modified so as to avoid exploring branches that are cyclic permutations of others already fathomed. The resulting implementation permits the solution of problems of realistic size within reasonable computational effort. Several examples are used to illustrate the applicability of the algorithm.     

Tabu search for a class of scheduling problems

Scheduling problems are often modeled as resourceconstrained problems in which critical resource assignments to tasks are known and the best assignment of resource time must be made subject to these constraints. Generalization toresource scheduling, where resource assignments are chosen concurrently with times results is a problem which is much more difficult. A simplified model of the general resource scheduling model is possible, however, in which tasks must be assigned a singleprimary resource, subject to constraints resulting from preassignment ofsecondary, or auxiliary, resources. This paper describes extensions and enhancements of tabu search for the special case of the resource scheduling problem described above. The class of problems is further restricted to those where it is reasonable to enumerate both feasible time and primary resource assignments. Potential applications include shift oriented production and manpower scheduling problems as well as course scheduling where classrooms (instructors) are primary and instructors (rooms) and students are secondary resources. The underlying model is a type of quadratic multiple choice problem which we call multiple choice quadratic vertex packing (MCQVP). Results for strategic oscillation and biased candidate sampling strategies are shown for reasonably sized real and randomly generated, synthetic, problem instances. The strategies are compared with other variations using consistent measures of solution time and quality developed for this study.     

Mathematical programming modeling of the Response Time Variability Problem

The Response Time Variability Problem (RTVP) is a scheduling problem that has recently been defined in the literature. The RTVP has a broad range of real-life applications from manufacturing to services and information technology. A previous study developed a position exchange heuristic to apply to initial sequences for the RTVP, and a MILP (Mixed Integer Linear Programming) to obtain optimal solutions with a practical limit of 25 units to be scheduled. This paper aims to improve the best mathematical programming model developed thus far in order to solve larger instances up to 40 units to optimality. The contribution of this paper is 4-fold: (i) larger instances can be solved to optimality by the off the shelf standard software; (ii) the new optimal solutions of the RTVP can be used to compare the results of heuristic procedures; (iii) the importance of modeling is demonstrated, as well as the huge impact that reformulation, redundant constraints and the elimination of symmetries have on the efficiency of MILPs is clearly established; finally (iv) a challenge to develop a customized optimization algorithm to rival the MILP solution efficiency for the RTVP is put forward.     

Mixed Integer Linear Programming in Process Scheduling: Modeling,Algorithms, and Applications

This paper reviews the advances of mixed-integer linear programming (MILP) based approaches for the scheduling of chemical processing systems. We focus on the short-term scheduling of general network represented processes. First, the various mathematical models that have been proposed in the literature are classified mainly based on the time representation. Discrete-time and continuous-time models are presented along with their strengths and limitations. Several classes of approaches for improving the computational efficiency in the solution of MILP problems are discussed. Furthermore, a summary of computational experiences and applications is provided. The paper concludes with perspectives on future research directions for MILP based process scheduling technologies.     

Scheduling Malleable Tasks on Parallel Processors to Minimize the Makespan

The problem of optimal scheduling n tasks in a parallel processor system is studied. The tasks are malleable, i.e., a task may be executed by several processors simultaneously and the processing speed of a task is a nonlinear function of the number of processors allocated to it. The total number of processors is m and it is an upper bound on the number of processors that can be used by all the tasks simultaneously. It is assumed that the number of processors is sufficient to process all the tasks simultaneously, i.e. n≤m. The objective is to find a task schedule and a processor allocation such that the overall task completion time, i.e. the makespan, is minimized. The problem is motivated by real-life applications of parallel computer systems in scientific computing of highly parallelizable tasks. An O(n) algorithm is presented to solve this problem when all the processing speed functions are convex. If these functions are all concave and the number of tasks is a constant, the problem can be solved in polynomial time. A relaxed problem, in which the number of processors allocated to each task is not required to be integer, can be solved in O(nmax {m,nlog ² m}) time. It is proved that the minimum makespan values for the original and relaxed problems coincide. For n=2 or n=3, an optimal solution for the relaxed problem can be converted into an optimal solution for the original problem in a constant time.     

Generalized cover facet inequalities for the generalized assignment problem

The generalized assignment problem is that of finding an optimal assignment of agents to tasks, where each agent may be assigned multiple tasks and each task is performed exactly once. This is an NP-complete problem. Algorithms that employ information about the polyhedral structure of the associated polytope are typically more effective for large instances than those that ignore the structure. A class of generalized cover facet-defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from generalized cover inequalities.     

Combinatorial integral approximation

We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process. The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms cplex on a numerical case study and hence further improves efficiency of our novel method.     

Multi-depot vehicle scheduling problems with time windows and waiting costs

The multi-depot vehicle scheduling problem with time windows (MDVSPTW) consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost. Each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots. The problem is formulated as an integer nonlinear multi-commodity network flow model with time variables and is solved using a column generation approach embedded in a branch-and-bound framework. This paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times. This new and more realistic cost structure gives rise to a nonlinear objective function in the model. Optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem (UBSP) and freight transport scheduling problem (FTSP). The results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used, and can efficiently treat the case with exact waiting costs.     

Strategic workforce planning for the Australian Defence Force

This paper presents a mixed integer linear programming (MILP) model to aid in planning a large, hierarchically structured workforce. The workforce (eg, all Army commissioned officers) is classified by occupational group and rank and modelled in yearly intervals. Personnel and their movements are modelled as stocks and flows, subject to constraints representing employment conditions and resource limitations. The task is to estimate personnel numbers and flows so as to minimize a composite cost function. The model’s detailed fidelity to actual conditions, and hence its value in practice, clearly exceed previous efforts in the workforce planning domain. To overcome computational challenges that arose when an MILP solver was applied directly, the authors developed an iterative solution approach, which has yielded an attractive combination of solution quality and computational performance.     

Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks

This paper deals with the branch and bound solution of process synthesis problems that are modelled as mixed-integer linear programming (MILP) problems. The symbolic integration of logic relations between potential units in a process network is proposed in the LP based branch and bound method to expedite the search for the optimal solution. The objective of this integration is to reduce the number of nodes that must be enumerated by using the logic to decide on the branching of variables and to determine by symbolic inference whether additional variables can be fixed at each node. The important feature of this approach is that it does not require additional constraints in the MILP and the logic can be systematically generated for process networks. Strategies for performing the integration are proposed that use the disjunctive and conjunctive normal form representations of the logic, respectively. Computational results will be presented to illustrate that substantial savings can be achieved.     

Finished-vehicle transporter routing problem solved by loading pattern discovery

This work addresses a new transportation problem in outbound logistics in the automobile industry: the finished-vehicle transporter routing problem (FVTRP). The FVTRP is a practical routing problem with loading constraints, and it assumes that dealers have deterministic demands for finished vehicles that have three-dimensional irregular shapes. The problem solution will identify optimal routes while satisfying demands. In terms of complex packing, finished vehicles are not directly loaded into the spaces of transporters; instead, loading patterns matching finished vehicles with transporters are identified first by mining successful loading records through virtual and manual loading test procedures, such that the packing problem is practically solved with the help of a procedure to discover loading patterns. This work proposes a mixed-integer linear programming (MILP) model for the FVTRP considering loading patterns. As a special class of routing models, the FVTRP is typically difficult to solve within a manageable computing time. Thus, an evolutionary algorithm is designed to solve the FVTRP. Comparisons of the proposed algorithm and a commercial MILP solver demonstrate that the proposed algorithm is more effective in solving medium- and large-scale problems. The proposed scheme for addressing the FVTRP is illustrated with an example and tested with benchmark instances that are derived from well-studied vehicle routing datasets.     

Globally solving a nonlinear UAV task assignment problem by stochastic and deterministic optimization approaches

In this paper, we consider a task allocation model that consists of assigning a set of m unmanned aerial vehicles (UAVs) to a set of n tasks in an optimal way. The optimality is quantified by target scores. The mission is to maximize the target score while satisfying capacity constraints of both the UAVs and the tasks. This problem is known to be NP-hard. Existing algorithms are not suitable for the large scale setting. Scalability and robustness are recognized as two main issues. We deal with these issues by two optimization approaches. The first approach is the Cross-Entropy (CE) method, a generic and practical tool of stochastic optimization for solving NP-hard problem. The second one is Branch and Bound algorithm, an efficient classical tool of global deterministic optimization. The numerical results show the efficiency of our approaches, in particular the CE method for very large scale setting.     

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.     

Generalized parallel-server fork-join queues with dynamic task scheduling

This paper introduces a generalization of the classical parallel-server fork-join queueing system in which arriving customers fork into multiple tasks, every task is uniquely assigned to one of the set of single-server queues, and each task consists of multiple iterations of different stages of execution, including task vacations and communication among sibling tasks. Several classes of dynamic polices are considered for scheduling multiple tasks at each of the single-server queues to maintain effective server utilization. The paper presents an exact matrix-analytic analysis of generalized parallel-server fork-join queueing systems, for small instances of the stochastic model, and presents an approximate matrix-analytic analysis and fixed-point solution, for larger instances of the model.     

Robust scheduling of parallel machines with sequence-dependent set-up costs

In this paper we propose a robust approach for solving the scheduling problem of parallel machines with sequence-dependent set-up costs. In the literature, several mathematical models and solution methods have been proposed to solve such scheduling problems, but most of which are based on the strong assumption that input data are known in a deterministic way. In this paper, a fuzzy mathematical programming model is formulated by taking into account the uncertainty in processing times to provide the optimal solution as a trade-off between total set-up cost and robustness in demand satisfaction. The proposed approach requires the solution of a non-linear mixed integer programming (NLMIP), that can be formulated as an equivalent mixed integer linear programming (MILP) model. The resulting MILP model in real applications could be intractable due to its NP-hardness. Therefore, we propose a solution method technique, based on the solution of an approximated model, whose dimension is remarkably reduced with respect to the original counterpart. Numerical experiments conducted on the basis of data taken from a real application show that the average deviation of the reduced model solution over the optimum is less than 1.5%.     

Approximations for the Two-Machine Cross-Docking Flow Shop Problem

We consider in this article the Two-Machine Cross-Docking Flow Shop Problem, which is a special case of scheduling with typed tasks, where we have two types of tasks and one machine per type. Precedence constraints exist between tasks, but only from a task of the first type to a task of the second type. The precedence relation is thus a directed bipartite graph. Minimizing the makespan is strongly NP-hard even with unit processing times, but any greedy method yields a 2-approximation solution. In this paper, we are interested in establishing new approximability results for this problem. More specifically, we investigate three directions: list scheduling algorithms based on the relaxation of the resources, the decomposition of the problem according to the connected components of the precedence graph, and finally the search of the induced balanced subgraph with a bounded degree.