In 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 Stinson et al. (1978). A tree search algorithm, based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature.Project Scheduling, Branch-and-Bound Methods, Networks/Graphs, Lower Bounds

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The Traveling Salesman Problem with Time Window and Precedence Constraints (TSP-TWPC) is to find an Hamiltonian tour of minimum cost in a graph G = (X, A) of n vertices, starting at vertex 1, visiting each vertex i ∈ X during its time window and after having visited every vertex that must precede i, and returning to vertex 1. The TSP-TWPC is known to be NP-hard and has applications in many sequencing and distribution problems. In this paper we describe an exact algorithm to solve the problem that is based on dynamic programming and makes use of bounding functions to reduce the state space graph. These functions are obtained by means of a new technique that is a generalization of the “State Space Relaxation” for dynamic programming introduced by Christofides et al. (Christofides, N., A. Mingozzi, P. Toth. 1981b. State space relaxation for the computation of bounds to routing problems. Networks 11 145–164.). Computational results are given for randomly generated test problems.

In this paper the n/I/q 2 O / x wjCj problem under the assumptions of nonpreemptive sequencing and sequence independent processing times is investigated. After pointing out the fundamental properties, some dominance sufficient conditions among sequences are obtained and a branch and bound algorithm is proposed. Computational results are reported and discussed.i

In this article we consider the problem of minimizing the maximum completion time of a sequence of n jobs on a single machine. Nonzero ready times and sequence‐dependent processing times are allowed. Upper bounds, lower bounds, and dominance criteria are proposed and exploited in a branch‐and‐bound algorithm. Computational results are given.