Flow Polyhedra and Resource Constrained Project Scheduling Problems (original) (raw)
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INFORMS Journal on Computing, 2005
W e propose a cooperation method between constraint programming and integer programming to compute lower bounds for the resource-constrained project scheduling problem (RCPSP). The lower bounds are evaluated through linear-programming (LP) relaxations of two different integer linear formulations. Efficient resource-constraint propagation algorithms serve as a preprocessing technique for these relaxations. The originality of our approach is to use additionally some deductions performed by constraint propagation, and particularly by the shaving technique, to derive new cutting planes that strengthen the linear programs. Such new valid linear inequalities are given in this paper, as well as a computational analysis of our approach. Through this analysis, we also compare the two considered linear formulations for the RCPSP and confirm the efficiency of lower bounds computed in a destructive way.
Strong Bounds for Resource Constrained Project Scheduling: Preprocessing and Cutting Planes
Comput. Oper. Res., 2020
Resource Constrained Project Scheduling Problems (RCPSPs) without preemption are well-known N P -hard combinatorial optimization problems. A feasible RCPSP solution consists of a time-ordered schedule of jobs with corresponding execution modes, respecting precedence and resources constraints. In this paper, we propose a cutting plane algorithm to separate five different cut families, as well as a new preprocessing routine to strengthen resource-related constraints. New lifted versions of the well-known precedence and cover inequalities are employed. At each iteration, a dense conflict graph is built considering feasibility and optimality conditions to separate cliques, odd-holes and strengthened Chvatal-Gomory cuts. The proposed strategies considerably improve the linear relaxation bounds, allowing a state-of-the-art mixed-integer linear programming solver to find provably optimal solutions for 754 previously open instances of different variants of the RCPSPs, which was not possible...
Project scheduling with resource constraints: A branch and bound approach
European Journal of Operational Research, 1987
This paper describes a branch and bound algorithm for project scheduling with resource constraints. The algorithm is 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 isbased on a Lagrangean relaxation of the formulation. This last relaxation involves a problem which is $ 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.
Resource-constrained project scheduling
Abstract: Resource-constrained project scheduling involves the scheduling of project activities subject to precedence and resource constraints in order to meet the objective(s) in the best possible way. The area covers a wide variety of problem types. The objective of this paper is to provide a survey of what we believe are important recent in the area . Our main focus will be on the recent progress made in and the encouraging computational experience gained with the use of optimal solution procedures for the basic resource-constrained project scheduling problem (RCPSP) and important extensions. The RCPSP involves the scheduling of a project its duration subject to zero-lag finish-start precedence constraints of the PERT/CPM type and constant availability constraints on the required set of renewable resources. We discuss recent striking advances in dealing with this problem using a new depth-first branch-and-bound procedure, elaborating on the effective and efficient branching schem...
In a previous paper (De , we presented 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 branchand-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 . A comparison with other computational results reported in the literature is included.
Time-constrained project scheduling
Journal of Scheduling, 2008
We propose a new approach for scheduling with strict deadlines and apply this approach to the Time-Constrained Project Scheduling Problem (TCPSP). To be able to meet these deadlines, it is possible to work in overtime or hire additional capacity in regular time or overtime. For this problem, we develop a two stage heuristic. The key of the approach lies in the first stage in which we construct partial schedules. In these partial schedules, jobs may be scheduled for a shorter duration than required. The second stage uses an ILP formulation of the problem to turn a partial schedule into a feasible schedule, and to perform a neighborhood search. The developed heuristic is quite flexible and, therefore, suitable for practice. We present experimental results on modified RCPSP benchmark instances. The two stage heuristic solves many instances to optimality, and if we substantially decrease the deadline, the rise in cost is only small.
More Efficient Algorithms for Constrained Project Scheduling Problems
In an earlier paper, we have described a framework to compute a destructive lower bound for a number of resource constrained project scheduling (RCPS) problems, which is based on column generation. If a certain threshold cannot be rejected, then we attempt finding a feasible solution with this value by solving an ILP. In this paper, we present two methods to speed up the computation. The first one is that we shrink the timewindows for the jobs by deriving additional release dates and deadlines from the solution that we obtained when determining the lower bound. The second method is to enforce the exact precedence delays by adding additional equalities. Our computational experiments show that each method can reduce the computation time by a factor of at most 10.
I n this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by .
On Project Scheduling with And/Or Precedence Constraints
We address a PERT/CPM scheduling problem with AND/OR constraints and examine its relations with extremal problems in grammars and hypergraphs. We demonstrate that two scheduling algorithms developed independently by and have different worst-case complexity and, in a sense, are incomparable.
Resource-constrained project scheduling: A survey of recent developments
Computers & Operations Research, 1998
Resource-constrained project scheduling involves the scheduling of project activities subject to precedence and resource constraints in order to meet the objective(s) in the best possible way. The area covers a wide variety of problem types. The objective of this paper is to provide a survey of what we believe are important recent developments in the area. Our main focus will be on the recent progress made in and the encouraging computational experience gained with the use of optimal solution procedures for the basic resource-constrained project scheduling problem (RCPSP) and important extensions.