zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Parallel machine scheduling with a convex resource consumption function. (English) Zbl 1125.90023
Summary: We consider some problems of scheduling jobs on identical parallel machines where job-processing times are controllable through the allocation of a nonrenewable common limited resource. The objective is to assign the jobs to the machines, to sequence the jobs on each machine and to allocate the resource so that the makespan or the sum of completion times is minimized. The optimization is done for both preemptive and nonpreemptive jobs. For the makespan problem with nonpreemptive jobs we apply the equivalent load method in order to allocate the resources, and thereby reduce the problem to a combinatorial one. The reduced problem is shown to be NP-hard. If preemptive jobs are allowed, the makespan problem is shown to be solvable in $O(n^2)$ time. Some special cases of this problem with precedence constraints are presented and the problem of minimizing the sum of completion times is shown to be solvable in $O(n\log n)$ time.

90B35Scheduling theory, deterministic
90C60Abstract computational complexity for mathematical programming problems
Full Text: DOI
[1] Armstrong, R.; Gu, S.; Lei, L.: An $O(Nlog(1/{\epsilon}))$ algorithm for the two-resource allocation problem with a non-differentiable convex objective function. Journal of the operational research society 46, 116-122 (1995) · Zbl 0835.90064
[2] Armstrong, R.; Gu, S.; Lei, L.: Solving a class of two-resource allocation problem by equivalent load method. Journal of the operational research society 48, 818-825 (1997) · Zbl 0890.90088
[3] Cheng, T. C. E.; Kovalyov, M. Y.: Single machine batch scheduling with deadlines and resource dependent processing times. Operations research letters 17, 243-249 (1995) · Zbl 0858.90073
[4] Cheng, T. C. E.; Janiak, A.: A permutation flow-shop scheduling problem with convex models of operation processing times. Annals of operations research 96, 39-60 (2000) · Zbl 0997.90035
[5] Cheng, T. C. E.; Janiak, A.; Kovalyov, M. Y.: Bicriterion single machine scheduling with resource dependent processing times. SIAM journal on optimization 8, No. 2, 617-630 (1998) · Zbl 0907.68113
[6] Conway, W. L.; Maxwell, W. L.; Miller, L. W.: Theory of scheduling. (1967) · Zbl 1058.90500
[7] Daniels, R. L.; Sarin, R. K.: Single machine scheduling with controllable processing times and number of jobs tardy. Operations research 37, No. 6, 981-984 (1989) · Zbl 0686.90023
[8] Daniels, R. L.: A multi-objective approach to resource allocation in single machine scheduling. European journal of operational research 48, 226-241 (1990) · Zbl 0825.90535
[9] Graham, R. L.; Lawler, E. L.; Lenstra, J. K.; Kan, A. H. G. Rinnooy: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of discrete mathematics 3, 287-326 (1979) · Zbl 0411.90044
[10] Janiak, A.: One-machine scheduling with allocation of continuously-divisible resource and with no precedence constraints. Kybernetika 23, 289-293 (1987) · Zbl 0635.90048
[11] Janiak, A.: Minimization of the resource consumption under a given deadline in the two-processor flow-shop scheduling problem. Information processing letters 32, 101-112 (1989) · Zbl 0685.90054
[12] Janiak, A.: Single machine scheduling problem with a common deadline and resource dependent release dates. European journal of operational research 53, 317-325 (1991) · Zbl 0743.90066
[13] Janiak, A.: Minimization of the makespan in a two-machine problem under given resource constraints. European journal of operational research 107, 325-337 (1998) · Zbl 0943.90031
[14] Janiak, A.; Szkodny, T.: Job-shop scheduling with convex models of operations. Mathematical and computer modeling 20, No. 2, 59-68 (1994) · Zbl 0810.90069
[15] Janiak, A.; Kovalyov, M. Y.: Single machine scheduling subject to deadlines and resource dependent processing times. European journal of operational research 94, 284-291 (1996) · Zbl 0947.90584
[16] Kaspi, M.; Shabtay, D.: Convex resource allocation for minimizing the makespan in a single machine with release dates. Computers and operations research 31, No. 9, 1481-1489 (2004) · Zbl 1076.68018
[17] Karush, W., 1939. Minima of functions of several variables with inequalities as side conditions. MS Thesis, Department of Mathematics, University of Chicago.
[18] Kuhn, H. W.; Tucker, A. W.: Nonlinear programming. Proceedings of the second Berkeley symposium on mathematical statistics and probability, 481-492 (1951) · Zbl 0044.05903
[19] Li, C. L.: Scheduling with resource dependent release dates--a comparison of two different resource consumption functions. Naval research logistics 41, 807-819 (1994) · Zbl 0819.90045
[20] Mcnaughton, R.: Scheduling with deadlines and loss functions. Management science 6, 1-12 (1959) · Zbl 1047.90504
[21] Monma, C. L.; Schrijver, A.; Todd, M. J.; Wei, V. K.: Convex resource allocation problems on directed acyclic graphs: duality, complexity, special cases and extensions. Mathematics of operations research 15, 736-748 (1990) · Zbl 0717.90080
[22] Scoot, S. C.; Jefferson, T. R.: Allocation of resources in project management. International journal of systems science 26, No. 2, 413-420 (1995) · Zbl 0821.90069
[23] Van Wassenhove, L.; Baker, K. R.: A bicriterion approach to time/cost trade-offs in sequencing. European journal of operational research 11, 48-54 (1982) · Zbl 0482.90043
[24] Vickson, R. G.: Two single machine sequencing problems involving controllable job processing times. AIIE transactions 12, No. 3, 258-262 (1980)
[25] Williams, T. J.: Analysis and design of hierarchical control systems with special reference to steel plant operations. (1985)