4 research outputs found
A Single-Machine Scheduling Problem with Uncertainty in Processing Times and Outsourcing Costs
We consider a single-machine scheduling problem with an outsourcing option in an environment where the processing time and outsourcing cost are uncertain. The performance measure is the total cost of processing some jobs in-house and outsourcing the rest. The cost of processing in-house jobs is measured as the total weighted completion time, which can be considered the operating cost. The uncertainty is described through either an interval or a discrete scenario. The objective is to minimize the maximum deviation from the optimal cost of each scenario. Since the deterministic version is known to be NP-hard, we focus on two special cases, one in which all jobs have identical weights and the other in which all jobs have identical processing times. We analyze the computational complexity of each case and present the conditions that make them polynomially solvable
A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs
This work deals with a class of problems under interval data uncertainty,
namely interval robust-hard problems, composed of interval data min-max regret
generalizations of classical NP-hard combinatorial problems modeled as 0-1
integer linear programming problems. These problems are more challenging than
other interval data min-max regret problems, as solely computing the cost of
any feasible solution requires solving an instance of an NP-hard problem. The
state-of-the-art exact algorithms in the literature are based on the generation
of a possibly exponential number of cuts. As each cut separation involves the
resolution of an NP-hard classical optimization problem, the size of the
instances that can be solved efficiently is relatively small. To smooth this
issue, we present a modeling technique for interval robust-hard problems in the
context of a heuristic framework. The heuristic obtains feasible solutions by
exploring dual information of a linearly relaxed model associated with the
classical optimization problem counterpart. Computational experiments for
interval data min-max regret versions of the restricted shortest path problem
and the set covering problem show that our heuristic is able to find optimal or
near-optimal solutions and also improves the primal bounds obtained by a
state-of-the-art exact algorithm and a 2-approximation procedure for interval
data min-max regret problems
Database query optimisation based on measures of regret
The query optimiser in a database management system (DBMS) is responsible for
�nding a good order in which to execute the operators in a given query. However, in
practice the query optimiser does not usually guarantee to �nd the best plan. This is
often due to the non-availability of precise statistical data or inaccurate assumptions
made by the optimiser. In this thesis we propose a robust approach to logical query
optimisation that takes into account the unreliability in database statistics during
the optimisation process. In particular, we study the ordering problem for selection
operators and for join operators, where selectivities are modelled as intervals rather
than exact values. As a measure of optimality, we use a concept from decision theory
called minmax regret optimisation (MRO).
When using interval selectivities, the decision problem for selection operator ordering
turns out to be NP-hard. After investigating properties of the problem and
identifying special cases which can be solved in polynomial time, we develop a novel
heuristic for solving the general selection ordering problem in polynomial time. Experimental
evaluation of the heuristic using synthetic data, the Star Schema Benchmark
and real-world data sets shows that it outperforms other heuristics (which take
an optimistic, pessimistic or midpoint approach) and also produces plans whose regret
is on average very close to optimal.
The general join ordering problem is known to be NP-hard, even for exact selectivities.
So, for interval selectivities, we restrict our investigation to sets of join
operators which form a chain and to plans that correspond to left-deep join trees.
We investigate properties of the problem and use these, along with ideas from the
selection ordering heuristic and other algorithms in the literature, to develop a
polynomial-time heuristic tailored for the join ordering problem. Experimental evaluation
of the heuristic shows that, once again, it performs better than the optimistic,
pessimistic and midpoint heuristics. In addition, the results show that the heuristic
produces plans whose regret is on average even closer to the optimal than for
selection ordering