Computational experience with a branch-and-bound procedure for the resource-constrained project scheduling problem with generalized precedence relations.

In a previous paper (De Reyck and Herroelen, 1996a), we presented an optimal procedure for the resource-constrained project scheduling problem (RCPSP) with generalised precedence relations (further denoted as RCPSP-GPR) with the objective of minimizing the project makespan. The RCPSP-GPR extends the RCPSP to arbitrary minimal and maximal time lags between the starting and completion times of activities. The procedure is a depth-first branch -and-bound algorithm in which the nodes in the search tree represent the original project network extended with extra precedence relations, which resolve a resource conflict present in the project network of the parent node. Resource conflicts are resolved using the concept of minimal delaying alternatives, i.e. minimal sets of activities which, when delayed, release enough resources to resolve the conflict. Precedence- and resource-based lower bounds as well as dominance rules are used to fathom large portions of the search tree. In this paper we report new computational experience with the algorithm using a new RCPSP-GPR random problem generator developed by Schwindt (1995). A comparison with other computational results reported in the literature is included.Scheduling;

An optimal procedure for the unconstrained max-NPV project scheduling problem with generalized precedence relations.

The unconstrained max-npv project scheduling problem involves the scheduling of the activities of a project in order to maximize its net present value. Assume a project represented in activity-on-mode (AoN) notation, in which the activities have a known duration and are subject to technological precedence constraints. Throughout each activity, a series of cash outflows and receipts may occur, which allows for the computation of a terminal cash flow value (positive or negative) upon the completion. The project is to be scheduled against a fixed deadline in the absence of resource constraints. Several procedures have been presented in the literature to cope with this problem. In this paper, we describe how one of the most efficient optimal procedures can be adapted to cope with generalized precedence relations, which introduce arbitrary minimal and maximal time lags between the start and completion of activities. The procedure has been programmed in Microsoft° Visual ++ 2.0 under Windows NT for use on a personal computer. Extensive computational results are reported.Scheduling; Optimal;

Phase transitions in project scheduling.

The analysis of the complexity of combinatorial optimization problems has led to the distinction between problems which are solvable in a polynomially bounded amount of time (classified in P) and problems which are not (classified in NP). This implies that the problems in NP are hard to solve whereas the problems in P are not. However, this analysis is based on worst-case scenarios. The fact that a decision problem is shown to be NP-complete or the fact that an optimization problem is shown to be NP-hard implies that, in the worst case, solving it is very hard. Recent computational results obtained with a well known NP-hard problem, namely the resource-constrained project scheduling problem, indicate that many instances are actually easy to solve. These results are in line with those recently obtained by researchers in the area of artificial intelligence, which show that many NP-complete problemsexhibit so-called phase transitions, resulting in a sudden and dramatic change of computational complexity based on one or more order parameters that are characteristic of the system as a whole. In this paper we provide evidence for the existence of phase transitions in various resource-constrained project scheduling problems. We discuss the use of network complexity measures and resource parameters as potential order parameters. We show that while the network complexity measures seem to reveal continuous easy-hard or hard-easy phase-transitions, the resource parameters exhibit an easy-hard-easy transition behaviour.Networks; Problems; Scheduling; Algorithms;

An optimal procedure for the resource-constrained project scheduling problem with discounted cash flows and generalized precedence relations.

In this paper, we study the resource-constrained project scheduling problem (RCPSP) with discounted cash flows and generalized precedence relations (further denoted as RCPSPDC-GPR). The RCPSPDC-GPR extends the RCPSP to (a) arbitrary minimal and maximal time lags between the starting and completion times of activities and (b) the non-regular objective function of maximizing the net present value of the project with positive and/or negative cash flows associated with the activities.). To the best of our knowledge, the literature on the RCPSPDC-GPR is completely void. We present a depth-first branch-and-bound algorithm in which the nodes in the search tree represent the original project network extended with extra precedence relations which resolve a number of resource conflicts. These conflicts are resolved using the concept of a minimal delaying mode (De Reyck and Herroelen, 1996b). An upper bound on the project net present value as well as several dominance rules are used to fathom large portions of the search tree. Extensive computational experience on a randomly generated benchmark problem set is obtained.Scheduling; Optimal; Discounted cash flow; Cash flow;

A branch-and-bound procedure for the resource-constrained project scheduling problem with generalized precedence relations.

We present an optimal procedure for the resource-constrained project scheduling problem (RCPSP) with generalized precedence relations (further denoted as RCPSP-GPR) with the objective of minimizing the project makespan. The RCPSP-GPR extends the RCPSP to arbitrary minimal and maximal time lags between the starting and completion times of activities. The procedure is a depth-first branch-and-bound algorithm in which the nodes in the search tree represent the original project network extended with extra precedence relations which resolve a resource conflict present in the parent node. Resource conflicts are resolved using the concept of minimal delaying alternatives, i.e. minimal sets of activities which, when delayed, release enough resources to resolve the conflict. Precedence and resource-based lower bounds as well as dominance rules are used to fathom large portions of the search tree. The procedure can be extended to other regular measures of performance by some minor modifications. Even non-regular measures of performance, such as the maximinization of the net present value of the project or resource levelling objectives, can be handled. The procedure has been programmed in Microsoft* Visual C++ for use on a personal computer. Extensive computational experience is obtained.Scheduling;

New computational results on the discrete time/cost trade-off probem in project networks.

We describe a new exact procedure for the discrete time/cost trade-off problem in deterministic activity-on-the-arc networks of the CPM type, where the duration of each activity is a discrete, nonincreasing function of the amount of a single resource committed to it. The objective is to construct the complete and efficient time/cost profile over the set of feasible project durations. The procedure uses a horizon-varying approach based on the iterative optimal solution of the problem of minimizing the sum of the resource use over all activities subject to the activity precedence constraints and a project dealine. This optimal solution is derived using a branch-a- bound procedure which computes lower bounds by making convex piecewise linear underestimations of the discrete time/cost trade-off curves of the activities to be used as an input for an adapted version of the Fulkerson labelling algorithm for the linear time/cost trade-off problem. Branching involves the selection of an acrivity in order to partition its set of execution modes into two subsets which are used to derive improved convex piecewise linear underestimations. The procedure has been programmed in Visual C++ under Windows NT and has been validated using a factorial experiment on a large set of problem instances.Networks; Problems; Scheduling; Time/cost trade-off problem; CPM; Optimal;

Algorithms for scheduling projects with generalized precedence relations.

Project scheduling under the assumption of renewable resource constraints and generalized precedence relations, i.e. arbitrary minimal and maximal time lags between the starting and completion times of the activities of the project, constitutes an important and challenging problem. Over the past few years considerable progress has been made in the use of exact solution procedure for this problem type and its variants. We review the fundamental logic and report new computational experience with a branch-and-bound procedure for optimally solving resource-constrained project scheduling problems with generalized precedence relations of the precedence diagramming type, i.e. start-start, start-finish, finish-start and finish-finish relations with minimal time lags for minimizing the project makespan. Subsequently, we review and report new results for several branch-and -bound procedures for the case of generalized precedence relations, including both minimal and maximal time lags, and demonstrate how the solution methodology can be expected to cope with other regular and nonregular objective functions such a smaximizing the net present value of a project.Networks; Problems; Scheduling; Algorithms; Functions; Net present value;

The Usefulness of Molecular Markers Approach for Developing Heterotic Groups in Maize

The phenomenon of heterosis provides a greatopportunity for plant breeders to gain greaterperformance and yield in the hybrids compared to theirinbred line parents. In most cross-pollinated crops likemaize, heterosis plays an important role as theperformance of the hybrids is of a great value. Heterosisgain much interest until recently and current studies stillattempt to elucidateone of these is utilizing molecularmarkers to estimate genetic distances among inbredlines followed by developing putative groups. In a welldened heterotic group, between-groups mating areexpected to produce better hybrids than within-groupsmating. Some studies of marker-aided heterotic groupdevelopment resulted in functional heterotic groups;while some other reported that the groups failed toprovide evidence of heterosis in the hybrids.Considering the prediction of hybrids' performance willdepend on genetic background of inbred lines, andmarkers technology are continuously improved, there isstill a good opportunity to obtain useful heterotic groupsfor a particular breeding population.Keywords: maize breeding, genetic distance, heterosis,molecular markers