Abstract
This paper considers the resource-constrained activity insertion problem with minimum and maximum time lags. The problem involves inserting a single activity in a partial schedule while preserving its structure represented through resource flow networks and minimizing the makespan increase caused by the insertion. In the general case, we show that finding a feasible insertion that minimizes the project duration is NP-hard. When only minimum time lags are considered and when activity durations are strictly positive, we show that the problem is polynomially solvable, generalizing previously established results on activity insertion for the standard resource-constrained project scheduling problem.
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Abbreviations
- RCPSP:
-
resource-constrained project scheduling problem
- RCPSP/Max:
-
RCPSP with minimum and maximum time lags
- RCAIP:
-
resource-constrained activity insertion problem
- n :
-
number of activities
- G(V,E,l):
-
activity-on-node graph
- V={A 0,…,A n+1}:
-
set of activities
- E :
-
set of precedence relations (A i ,A j )
- p i :
-
duration of activity A i
- l ij :
-
time lag for (A i ,A j )∈E
- m :
-
number of resources
- ℛ={R 1,…,R m }:
-
set of resources
- B k :
-
number of available units for resource R k
- b i,k :
-
number of units of R k required by A i
- S i :
-
start time of A i
- \(\mathcal{A}_{t}\) :
-
set of activities in process at time t
- (P):
-
short notation for the RCPSP/max problem
- (P −x ):
-
short notation for the RCPSP/max where b xk =0, ∀ R k ∈ℛ
- (P x ):
-
short notation for the RCAIP where A x has to be inserted
- δ i,j :
-
longest path from A i to A j in G(V,E,l)
- f i,j,k :
-
number of resource R k units transferred from A i to A j
- \({\mathcal{G}}(f)\) or \({\mathcal{G}}\) :
-
graph induced by flow f
- F(f) or F :
-
set of arcs (precedence constraints) induced by flow f
- L ij (f) or L ij :
-
weight of arc (A i ,A j ) induced by flow f
- Δ i,j (f) or Δ i,j :
-
longest path from A i to A j in \({\mathcal{G}}(f)\)
- q i,j,k :
-
part of flow f i,j,k rerouted to the inserted activity
- (α,β):
-
ordered pair of set of activities representing an insertion
- α :
-
set of possible resource predecessors
- β :
-
set of possible resource successors
- Q k (α,β):
-
amount of R k units available for insertion in (α,β)
- \({\mathcal{G}}(\alpha,\beta)\) :
-
graph issued from the insertion of A x in (α,β)
- F(α,β):
-
set of arcs issued from the insertion of A x in (α,β)
- L i,j (α,β):
-
weight of arc (A i ,A j ) after insertion of A x in (α,β)
- Δ i,j (α,β):
-
longest path from A i to A j in \({\mathcal{G}}(\alpha,\beta)\)
- \(\mathcal{C}_{q}(\alpha,\beta)\) :
-
set of type q cycles in \({\mathcal{G}}(\alpha,\beta)\) (q=1,2,3)
- ℒ q (α,β):
-
length of the longest cycle in \({\mathcal{G}}(\alpha,\beta)\) (q=1,2,3)
- ℳ q (α,β):
-
length of the longest path of type q in \({\mathcal{G}}(\alpha,\beta)\) (q=1,2,3)
- γ :
-
set of non-dummy activities linked with A x by a synchronization constraint
- μ(α):
-
set of activities i∈α of largest Δ 0,i +p i
- ν(α):
-
set of activities i∈β of largest Δ i,n+1
- ν′(α):
-
subset of activities i∈ν such that Δ x,i =−∞
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Artigues, C., Briand, C. The resource-constrained activity insertion problem with minimum and maximum time lags. J Sched 12, 447–460 (2009). https://doi.org/10.1007/s10951-009-0124-x
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DOI: https://doi.org/10.1007/s10951-009-0124-x