By T'kindt V., Billaut J.-C.
Scheduling and multicriteria optimisation idea were topic, individually, to varied experiences. because the final two decades, multicriteria scheduling difficulties were topic to a starting to be curiosity. even if, a niche among multicriteria scheduling ways and multicriteria optimisation box exists. This publication is an try and acquire the effortless of multicriteria optimisation conception and the elemental versions and algorithms of multicriteria scheduling. it really is composed of various illustrations, algorithms and examples which can aid the reader in figuring out the offered options.
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5. ALGORITHM EELl /* T is the set of jobs to schedule */ 5 = 0; n i=i For i = n down to 1 Do F = {Ji eT/Ji has no successor in T}; If (F = 0) Then I The problem is not feasible; End If; _ _ Let Jee Fhe such that fe{P) = min {fk{P)); /* Break ties by choosing the job with the greatest processing time */ S={Je}//S; T = T-{Jeh Ce=_P; P = P-pe', End For; Print 5; [Lawler, IQTäT Fig. 5. An optimal algorithm for the l\prec\fmax problem Moore's algorithm for the l|di|f/ problem Consider the problem where n jobs have to be scheduled on a single machine and each job Ji has a due date di.
Henceforth, we can informally derive that enumeration is at least as hard as counting, since when the enumeration is performed the counting of solutions can be polynomially done whilst the converse is false. We first focus on the complexity of counting problems before considering the complexity of enumeration problems. Let us define the counting problem, denoted by C, associated to an optimisation problem O as follows: Counting problem C • Input data, or instances, denoted by 7. The set of all the instances is denoted by Do, • Counting question: how many optimal solutions to the objective of problem O exist ?
We see that for the decision problem this Turing machine is equivalent to a "^o/t'e" procedure which returns true or false. The complexity of a decision problem thus depends on the "best" Turing machine which we can propose. The encoding scheme used influences equally the efiiciency of the proposable Turing machine. e. which does not pointlessly complicate the obtained grammar. For all reasonable encoding schemes, the proposable Turing machines are judged equivalent ([Garey and Johnson, 1979]).