A path-relinking metaheuristic for the resource levelling problem

This paper deals with project scheduling problem with resource levelling objective function where precedence relations among activities are prescribed. We develop a dedicated path-relinking metaheuristic algorithm to tackle this problem. Computational results on randomly generated test sets indicate the developed procedure is efficient and outperforms the best available metaheuristic algorithms in the literature.     

Cost-effective allocations of bounded and binary resources in polynomial time

The problem considered is that of the allocation of linear and discrete resources to concave activities with binary effectiveness values of 0 or 1. The objective is to maximize the ratio of the total return to an affine cost. Previously described algorithms are extended to include an upper bound on the quantity of any resource allocated to given disjoint sets of activities, and on the quantity allocated to any activity from given disjoint sets of resources. It is demonstrated that the algorithm solves the problem in polynomial time.     

Approximation algorithms for scheduling on multi-core processor with shared speedup resources

We consider a joint resource partition and scheduling problem. We are given $m$ identical cores and discrete resources of total size $k$. We need to partition the resources among these cores. A set of jobs must be processed non-preemptively on these cores after the resource partition. The processing time of a job on a core depends on the size of resources allocated to that corresponding core. The resource allocation scheme is static, i.e., we cannot change the amount of resources that was allocated to a core during the whole scheduling. Hassidim et al. (2013) investigated this problem with a general processing time function, i.e., the processing time of a job is an arbitrary function of the level of resources allocated to that core. They provided an algorithm with approximation ratio of 36. In this paper, we improve the approximation ratio to 8 by presenting a new resource partition scheme. Next, we consider a special model where the core’s speed is proportional to its allocated resource, then we present two algorithms with improved approximation ratios.     

Solving a class of two-resource allocation problem by equivalent load method

We present an efficient algorithm for solving a class of two-resource allocation problem defined on a series-parallel graph, where nodes represent tasks of a given project and arcs represent precedence relationships. Two separate workloads are associated with each task and the time to complete a workload is inversely proportional to the amount of resource allocated. The time to complete a task is the maximum of the times taken to complete the two workloads. The problem is to allocate the two resources across the project so as to minimize the project duration. The proposed algorithm is derived based on the Equivalent Load Method by Monma, Schrijver, Todd, and Wei for the single-resource allocation problem.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem     

Project scheduling with resource constraints: A branch and bound approach

This paper describes a branch and bound algorithm for project scheduling with resource constraints. The algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods. Four lower bounds are examined. The first is a simple lower bound based on longest path computations. The second and third bounds are derived from a relaxed integer programming formulation of the problem. The second bound is based on the Linear Programming relaxation with the addition of cutting planes, and the third bound is based on a Lagrangean relaxation of the formulation. This last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient, though not polynomial, algorithm is given. The fourth bound is based on the disjunctive arcs used to model the problem as a graph.We report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources.     

Multi-objectives Tabu Search based algorithm for progressive resource allocation

Military course of action planning involves time and space synchronization as well as resource and asset allocation. A mission could be seen as a defined set of logical ordered tasks with time and space constraints. The resources to task rules require that available assets should be allocated to each task. A combination of assets might be required to execute a given task. The couple (task, resources) constitutes an action. This problem is formulated as a multi-objectives scheduling and resource allocation problem. Any solution is assessed based on a number of conflicting and heterogeneous objectives. In fact, military planning officers should keep perfecting the plan based on the Commander’s criteria for success. The scheduling problem and resource allocation problem are considered as NP-Hard Problems [A. Guitouni, B. Urli, J.-M. Martel, Course of action planning: A project based modelling, Working Paper, Faculté des sciences de l’ administration, Université Laval, Québec, 2005]. This paper is concerned with the multi-objectives resource allocation problem. Our objective is to find adequate resource allocation for given courses of action schedule. To optimize this problem, this paper investigates non-exact solution methods, like meta-heuristic methods for finding potential efficient solutions. A progressive resource allocation methodology is proposed based on Tabu Search and multi-objectives concepts. This technique explores the search space so as to find a set of potential efficient solutions without aggregating the objectives into a single objective function. It is guided by the principle of maximizing the usage of any resource before considering a replacement resource. Thus, a given resource is allocated to the maximum number of tasks for a given courses of action schedule. A good allocation is a potential efficient solution. These solutions are retained by applying a combination of a dominance rule and a multi-criteria filtering method. The performance of the proposed Pareto-based approach is compared to two aggregation approaches: weighted-sum and the lexicographic techniques. The result shows that a Pareto-based approach is providing better solutions and allowing more flexibility to the decision-maker.     

Multi-mode resource constrained multi-project scheduling and resource portfolio problem

This paper introduces a multi-project problem environment which involves multiple projects with assigned due dates; activities that have alternative resource usage modes; a resource dedication policy that does not allow sharing of resources among projects throughout the planning horizon; and a total budget. Three issues arise when investigating this multi-project environment. First, the total budget should be distributed among different resource types to determine the general resource capacities, which correspond to the total amount for each renewable resource to be dedicated to the projects. With the general resource capacities at hand, the next issue is to determine the amounts of resources to be dedicated to the individual projects. The dedication of resources reduces the scheduling of the projects’ activities to a multi-mode resource constrained project scheduling problem (MRCPSP) for each individual project. Finally, the last issue is the efficient solution of the resulting MRCPSPs. In this paper, this multi-project environment is modeled in an integrated fashion and designated as the resource portfolio problem. A two-phase and a monolithic genetic algorithm are proposed as two solution approaches, each of which employs a new improvement move designated as the combinatorial auction for resource portfolio and the combinatorial auction for resource dedication. A computational study using test problems demonstrated the effectiveness of the solution approach proposed.     

Multi-objective Meta-heuristics for the Traveling Salesman Problem with Profits

We introduce and test a new approach for the bi-objective routing problem known as the traveling salesman problem with profits. This problem deals with the optimization of two conflicting objectives: the minimization of the tour length and the maximization of the collected profits. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined or one of the objectives has been treated as a constraint. The purpose of our study is to find solutions to this problem using the notion of Pareto optimality, i.e. by searching for efficient solutions and constructing an efficient frontier. We have developed an ejection chain local search and combined it with a multi-objective evolutionary algorithm which is used to generate diversified starting solutions in the objective space. We apply our hybrid meta-heuristic to synthetic data sets and demonstrate its effectiveness by comparing our results with a procedure that employs one of the best single-objective approaches.      

Minimizing total completion time and total deviation of job completion times from a restrictive due-date

We study a single-machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due-date. The due-date is assumed to be restrictive, i.e., it may be sufficiently small to have an impact on the optimal sequence. When more weight is allocated to total job completion times, the problem is shown to have a polynomial time solution. When more weight is allocated to total completion time deviations from the due-date, the problem is NP-hard in the ordinary sense. For the latter case, we introduce an efficient dynamic programming algorithm, which is shown numerically to perform well in all our tests.     

A hybrid scatter search for the discrete time/resource trade-off problem in project scheduling

We develop a heuristic procedure for solving the discrete time/resource trade-off problem in the field of project scheduling. In this problem, a project contains activities interrelated by finish-start-type precedence constraints with a time lag of zero, which require one or more constrained renewable resources. Each activity has a specified work content and can be performed in different modes, i.e. with different durations and resource requirements, as long as the required work content is met. The objective is to schedule each activity in one of its modes in order to minimize the project makespan. We use a scatter search algorithm to tackle this problem, using path relinking methodology as a solution combination method. Computational results on randomly generated problem sets are compared with the best available results indicating the efficiency of the proposed algorithm.     

Incremental allocation of a discrete and convex resource to concave activities

The problem considered is that of the allocation of a given quantity of a discrete resource to activities described by concave return functions. The resource-consumption corresponding to each allocation is described by a convex function of the quantity allocated. An incremental solution algorithm is given, which specializes to the algorithm of Fox if the resource is linear, and to an algorithm of Katoh, Ibaraki and Mine if the return functions are linear.     

Scheduling interfering job sets on parallel machines

We consider bicriteria scheduling on identical parallel machines in a nontraditional context: jobs belong to two disjoint sets, and each set has a different criterion to be minimized. The jobs are all available at time zero and have to be scheduled (non-preemptively) on m parallel machines. The goal is to generate the set of all non-dominated solutions, so the decision maker can evaluate the tradeoffs and choose the schedule to be implemented. We consider the case where, for one of the two sets, the criterion to be minimized is makespan while for the other the total completion time needs to be minimized. Given that the problem is NP-hard, we propose an iterative SPT–LPT–SPT heuristic and a bicriteria genetic algorithm for the problem. Both approaches are designed to exploit the problem structure and generate a set of non-dominated solutions. In the genetic algorithm we use a special encoding scheme and also a unique strategy – based on the properties of a non-dominated solution – to ensure that all parts of the non-dominated front are explored. The heuristic and the genetic algorithm are compared with a time-indexed integer programming formulation for small and large instances. Results indicate that the both the heuristic and the genetic algorithm provide high solution quality and are computationally efficient. The heuristics proposed also have the potential to be generalized for the problem of interfering job sets involving other bicriteria pairs.     

Multi-project scheduling with two-stage decomposition

We consider a non-preemptive, zero time lag multi-project scheduling problem with multiple modes and limited renewable and nonrenewable resources. A two-stage decomposition approach is adopted to formulate the problem as a hierarchy of 0-1 mathematical programming models. In stage one; each project is reduced to a macro-activity with macro-modes. The macro-activities are combined into a single macro-activity network over which the macro-activity scheduling problem (MP) is defined, where the objective is the maximization of the net present value with positive cash flows and the renewable resource requirements are time-dependent. An exact solution procedure and a genetic algorithm (GA) approach are proposed for solving the MP. A GA is also employed to generate an initial solution for the exact solution procedure. The first stage terminates with a post-processing procedure to distribute the remaining resource capacities. Using the start times and the resource profiles obtained in stage one, each project is scheduled in stage two for minimum makespan. Three new test problem sets are generated with 81, 84 and 27 problems each, and three different configurations of solution procedures are tested.     

Scheduling jobs with time-resource tradeoff via nonlinear programming

We consider a scheduling problem where the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The objective is to find a resource allocation and a schedule that minimizes the makespan. We explicitly allow for succinctly encodable time-resource tradeoff functions, which calls for mathematical programming techniques other than those that have been used before. Utilizing a (nonlinear) integer mathematical program, we obtain the first polynomial time approximation algorithm for the scheduling problem, with performance bound (3+ε) for any ε>0. Our approach relies on a fully polynomial time approximation scheme to solve the nonlinear mathematical programming relaxation. We also derive lower bounds for the approximation.     

An algorithm for the estimation of a regression function by continuous piecewise linear functions

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L ₂ risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.     

A branch-and-bound algorithm for the resource-constrained project scheduling problem

We describe a time-oriented branch-and-bound algorithm for the resource-constrained project scheduling problem which explores the set of active schedules by enumerating possible activity start times. The algorithm uses constraint-propagation techniques that exploit the temporal and resource constraints of the problem in order to reduce the search space. Computational experiments with large, systematically generated benchmark test sets, ranging in size from thirty to one hundred and twenty activities per problem instance, show that the algorithm scales well and is competitive with other exact solution approaches. The computational results show that the most difficult problems occur when scarce resource supply and the structure of the resource demand cause a problem to be highly disjunctive.     

A branch and bound algorithm to minimize total tardiness of jobs in a two identical-parallel-machine scheduling problem with a machine availability constraint

This research focuses on the problem of scheduling jobs on two identical parallel machines that are not continuously available with the objective of minimizing total tardiness. After processing a given number of jobs, each machine requires a preventive maintenance task, during which the machine cannot process jobs. We present dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as an upper bound obtained from a heuristic algorithm. Performance of the algorithm is evaluated through a series of computational experiments on randomly generated instances and results are reported.     

MineLib: a library of open pit mining problems

Similar to the mixed-integer programming library (MIPLIB), we present a library of publicly available test problem instances for three classical types of open pit mining problems: the ultimate pit limit problem and two variants of open pit production scheduling problems. The ultimate pit limit problem determines a set of notional three-dimensional blocks containing ore and/or waste material to extract to maximize value subject to geospatial precedence constraints. Open pit production scheduling problems seek to determine when, if ever, a block is extracted from an open pit mine. A typical objective is to maximize the net present value of the extracted ore; constraints include precedence and upper bounds on operational resource usage. Extensions of this problem can include (i) lower bounds on operational resource usage, (ii) the determination of whether a block is sent to a waste dump, i.e., discarded, or to a processing plant, i.e., to a facility that derives salable mineral from the block, (iii) average grade constraints at the processing plant, and (iv) inventories of extracted but unprocessed material. Although open pit mining problems have appeared in academic literature dating back to the 1960s, no standard representations exist, and there are no commonly available corresponding data sets. We describe some representative open pit mining problems, briefly mention related literature, and provide a library consisting of mathematical models and sets of instances, available on the Internet. We conclude with directions for use of this newly established mining library. The library serves not only as a suggestion of standard expressions of and available data for open pit mining problems, but also as encouragement for the development of increasingly sophisticated algorithms.     

Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities

The aim of this paper is to establish the stability of weak efficient, efficient and Henig proper efficient sets of a vector optimization problem, using quasiconvex and related functions. We establish the Kuratowski?CPainlevé set-convergence of the minimal solution sets of a family of perturbed problems to the corresponding minimal solution set of the vector problem, where the perturbations are performed on both the objective function and the feasible set. This convergence is established by using gamma convergence of the sequence of the perturbed objective functions and Kuratowski?CPainlevé set-convergence of the sequence of the perturbed feasible sets. The solution sets of the vector problem are characterized in terms of the solution sets of a scalar problem, where the scalarization function satisfies order preserving and order representing properties. This characterization is further used to establish the Kuratowski?CPainlevé set-convergence of the solution sets of a family of scalarized problems to the solution sets of the vector problem.