1. Single Machine Scheduling with a Non-renewable Financial Resource
Evgeny R. Gafarov
Alexander A. Lazarev
Frank Werner
Institute of Control Sciences
of the Russian Academy
of Sciences
Otto-von-Guericke
Universität Magdeburg

2. Definitions and literature review
Complexity results (with two proofs):
NP-hardness of 1|NR, pj=p|∑Tj
NP-hardness of 1|NR, dj=d, gj=g|∑Tj
Budget scheduling

3. Definitions
Single machine n jobs j=1,2,…,n pj processing time dj due date
Non-renewable resource (money, energy)
Set of times {t0, t1, …, ty} of earnings of the resource
Receipt of resource G(ti) at time ti
Consumption of resource gj when job j starts

4. Instance Receipt of money G(0)+G(20)+G(30) = $300 > $250 = g2 ti
Job j 1 2 3 gj
$100
$250
$10 pj 25 15 20
Receipt of money ti 20 30 50
G(ti)
$50
$200
$60 2 1 3 ti 20 30 50 95
G(0)+G(20)+G(30) = $300 > $250 = g2

5. Literature review
J. Carlier and A.H.G. Rinnooy Kan (1982) Scheduling subject to Nonrenewable-Resource Constraints.Oper. Res. Lett. 1(2), (precedence constraints, not a single machine)
R. Slowinski (1984) Preemptive Scheduling of Independent Jobs on Parallel Machines subject to Financial Constraints. European J. Oper. Res. 15,
A. Toker, S. Kondakci and N. Erkip (1991) Scheduling Under a Non-renewable Resource Constraint. J. Oper. Res. Soc. 42(9), (two machines, makespan, unit supply of a resource at each time period)
A. Janiak (1986) One-Machine Scheduling Problems with Resource Constraints. System Modelling and Optimization. Springer Berlin / Heidelberg, (single machine, processing times or the release times depend on the consumption of a non-renewable resource)

6. Complexity results (Total Tardiness)
Problem
Comp.results
Reduction from
1|NR|Cmax
NP-hard (strong sense)
3-Partition
1|NR|∑Uj
1|NR|∑Cj
1|rj|∑Cj
1|NR|Lmax
1|rj|Lmax
Complexity results (Total Tardiness)
Problem
NP hard special case
Reduction from
1|NR|∑Tj
dj=d (strong sense)
3-Partition
gj=g
1||∑Tj
pj=p
Partition
dj=d, gj=g

7. NP-hardness proof for 1|NR, pj=p|∑Tjti M
Set of on-time jobs

8. NP-hardness proof for 1|NR, pj=p|∑Tj
2n+1 jobs ti d

9. Budget scheduling
Non-renewable resource (money, energy)
Set of times {t0, t1, …, ty} of earnings of the resource
Receipt of resource G(ti) at time ti
Consumption of resource gj- when job j starts
(has the same meaning as gj)
Earnings of resource gj+ when job j ends
Financial scheduling – special case, when all gj+ = 0

10. Properties of feasible solutions.
Budget scheduling.
Properties of feasible solutions.

11. Thanks for attention Evgeny R. Gafarov Institute of Control Sciences
Alexander A. Lazarev
Frank Werner
Institute of Control Sciences
of the Russian Academy
of Sciences
Otto-von-Guericke
Universität Magdeburg