An Algorithm for the Discrete Time-Cost Tradeoff Problem Based on Lagrangean Relaxation

This paper deals with the discrete time-cost tradeoff problem (DTCTP) which is well-known to be NP-hard. We propose to compute lower bounds based on Lagrangean relaxation. The basic idea is to decompose the activity-on-node network into subnetworks by relaxing some precedence constraints. Partial solutions provided by the lower bounds are used also to compute upper bounds, that is feasible solutions. Finally, the lower and upper bounds are incorporated into a branch-and-bound algorithm. 1. The Discrete Time-Cost Tradeoff Problem Assume an acyclic activity-on-node network with V = {0, 1, ..., n, n+1} being the set of activities, where 0 is the source and n + 1 is the sink of the network, and V V E × ⊂ being the set of precedence constraints. For each activity V j ∈ a set j M of duration/cost pairs, so-called modes, is given. For each mode j M m ∈ , let 0 N jm p ∈ denote the processing time of j in mode m and let 0 ≥ ∈ R jm c denote the cost for performing activity j in mode m. Without loss of generality, we assume that j M m m ∈ ' , with m < m' implies that ' jm p jm p > and ' jm c jm c > . The problem is to find a mode assignment for the activities such that a given deadline T can be met with minimum cost while respecting all precedence constraints. A survey of related work on this problem can be found in [2]. Using the modes with the shortest processing times, it is straightforward to compute the earliest start time j ES , the earliest completion time j EC , and the latest completion time j LC for each activity j. Formally, the problem can be stated as follows with decision variables j C for the completion time of activity j, and jm x a binary mode indicator which is 1, if mode m is assigned to activity j, and 0, otherwise. ∑ ∑ ∈ ∈ V j M m jm jm j x c min ∑ ∈ = j M m jm x t s 1 . . V j ∈ ∑ ∈ ≥ − − j M m jm jm x p i C j C 0 E j i ∈ ) , ( T n C ≤ +1 0 ≥ j C V j ∈ } 1 , 0 { ∈ jm x j M m V j ∈ ∈ , 2. Lagrangean Relaxation The Lagrangean relaxation is based on the idea that the network is partitioned into smaller subnetworks (see [1]). Let H be the index set for these partitions and, for H h ∈ , V h V ⊆ be the set of nodes in subnetwork h, and E h V h V h E ⊆ × ⊆ be the set of arcs in the subnetwork h induced by h V . Let E be the set of arcs in E which have start and end nodes from different subnetworks. Since we assume a partition, we have V h V H h = ∈ U , = ′ I h V h V Ø, for h h ′ ≠ , and, E h E H h E = ∈ ∪ U . Now, let jm x and j C be feasible w.r.t. constraints h R , where