The proactive and reactive resource-constrained project scheduling problem : The crucial role of buer-based reactions M. Davari
E. Demeulemeester
KU Leuven Research group for operations management e-mail :
[email protected]
In our previous work [1], we formulated an integrated proactive and reactive scheduling problem with a combined cost function which includes a baseline schedule cost as well as costs of a series of reactions. Solutions to this problem are PR-policies. A PR-policy is described by a set of decision rules that dictate certain transitions among schedules. In our current work, we aim at understanding the importance of certain classes of reactions (i.e., the class of selection-based reactions and its subclass, the class of buer-based reactions) in constructing an optimal PR-policy. We are given a set N = {0, 1, ..., n+1} of activities where activities 0 and n+1 are the dummy start and dummy end activities. Each activity i ∈ N 0 = N \{0, n+ 1} has a stochastic non-negative integer duration p˜i , with pmin ≤ p˜i ≤ pmax , i i which follows a discrete distribution dist(˜ pi ). We assume that these stochastic durations are independently distributed. The vector p ˜ = (˜ p0 , p˜1 , ..., p˜n+1 ) can be represented by a nite supporting set P = {p1 , ..., p|p| } of realizations where each realization pl represents a vector of durations pl = (pl0 , pl1 , ..., pln+1 ) ∈ P. Notice that the durations of the dummy activities are not stochastic (p˜0 = p˜n+1 = 0). We are also given a set R of renewable resource types. Each job i requires rik units of resource type k ∈ R during its processing time and the resource availability of resource type k is denoted by Rk . The set E ∈ {(i, j)|i, j ∈ N } de nes precedence constraints among the activities where the pair (i, j) ∈ E indicates that activity j cannot be started before activity i is completed. P A set F S of activities is a forbidden set if E ∩ (F S × F S) = ∅ and ∃k ∈ R : i∈F S rik > Rk . A forbidden set F S is minimal if for every i ∈ F S , the set F S \ {i} is not a forbidden set. We de ne F (·) as the set of all minimal forbidden sets with · being a partial order among activities. One may use extra resource arcs to eliminate all minimal forbidden sets. Let us de ne selection X ⊂ N × N \ T (E) as a set of pairs where each pair represents a resource arc. We assume that X is a strict partial order on N (i.e. irre exible and transitive).
De nition 1 (Su cient selection) A selection
only if G(N, E ∪ X) is acyclic and F (E ∪ X) = ∅.
X
is called su cient if and
De nition 2 (Selection-based reaction) A reaction from schedule s to sche-
dule s0 is selection-based if there is a su cient selection X that is feasible for both s and s0 .
BB 34.90%
SNB 13.67%
BBO 88.92%
SNBO 9.95%
NSBO 1.13%
NSB 52.43% a The average contributions of dierent classes of reaction in the whole network
b The average contributions of dierent classes of reaction in the optimal PRpolicy
Figure
1 The average contributions of dierent classes of reaction for the setting where wb = 25 and wr = 0 Let ES(X, p) denote the induced early-start schedule for every given pair (X, p). A 3-tuple (X, p, p0 ) induces a pair of schedules (s, s0 ) if X is su cient, s = ES(X, p) and s0 = ES(X, p0 ).
De nition 3 (Buer-based reaction) A reaction from
to s0 is a buerbased reaction if there exists a 3-tuple (X, b, b ) that induces (s, s0 ). s
0
To understand the importance of the buer-based reactions, we depict in Figure 1 the contributions of three mutually exclusive and collectively exhaustive classes of reactions in the network and in its associated optimal PR-policy. The classes under comparison are the class of non-selection based (NSB) reactions, the class of buer-based (BB) reactions and the class of selection-but-not-buer-based (SNB) reactions. Figure 1a shows the contributions of these classes of reactions in the whole network and Figure 1b displays the contributions of these classes of reactions in the associated optimal PR-policy. The futility of non-selectionbased reactions in the optimal PR-policy is very clear in Figure 1 (NSBO represents the percentage of non-buer-based reactions in the optimal PR-policy) : although 52.43 percent of reactions are non-selection-based, the contribution of these reactions in the optimal PR-policy is only 1.13 percent. It is also clear that buer-based reactions are very important : as stated before, despite the fact that only 34.90 percent of reactions are buer-based, their contribution in the optimal PR-policy is very high (88.92 percent).
Références [1] Davari, M. and Demeulemeester, E. The proactive and reactive resourceconstrained project scheduling problem. Technical Report KBI_1613, KU Leuven, 2016.
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