Robust scheduling with budgeted uncertainty

Abstract

In this work we study min max robust scheduling problems assuming that the processing times can take any value in the budgeted uncertainty set introduced by Sim (2003,2004). We consider problems on a single machine that minimize the (weighted and unweighted) sum of completion times and problems that minimize the makespan on parallel and unrelated machines. We provide polynomial algorithms and approximation algorithms: constant factor, average non-constant factor, (fully or not) polynomial time approximation schemes. In addition, we prove that the robust version of minimizing the weighted completion time on a single machine is N P-hard in the strong sense.Keywords approximation algorithms, robust optimization, scheduling IntroductionScheduling is a very wide topic in combinatorial optimization with applications ranging from production and manufacturing systems to transportation and logistics systems. Stated generally, the objective of scheduling is to allocate optimally scarce resources to activities over time. The practical relevance and the difficulty of solving the general scheduling problem have motivated an intense research activity in a large variety of scheduling environments. Scheduling problems are usually defined in the following way: given a set of n jobs represented by J , a set of m machines represented by M, and processing times represented by the tuple p, we look for a schedule σ of the jobs on the machines that satisfies the side constraints, represented by the set S of feasible schedules, and minimize objective function f (σ, p). Formally, this amounts to solve optimization problem min σ∈S f (σ, p).Various sources of uncertainty affect real scheduling problems, among which machine breakdowns, working environment changes, worker performance instabilities, tool quality variations and unavailability. Ignoring these uncertainties usually yields schedules that perform poorly under real conditions. Hence, researchers have introduced frameworks where the uncertainty is directly taken into account either by considering random variables as input or in a worst-case approach where the uncertainty parameters are constrained in a set. These frameworks are respectively denoted by Stochastic Programming and Robust Optimization (RO). We disregard the former in this paper because of its requirement for a probabilistic distribution of the random inputs, which is very difficult to obtain in practice. We focus instead on Robust Scheduling, which models the uncertainty on the processing times by a finite set U ⊂ N n . 1 In the robust problem, the maximum value of f (σ, p) over all p ∈ U should be minimized. Formally, this amounts to solve optimization problem min σ∈S max p∈U f (σ, p), or equivalently, min σ∈S F (σ, U ) where F (σ, U ) = max p∈U f (σ, p) represents the robust objective function. We say that a schedule σ * ∈ S is robust if it solves the associated scheduling problem min σ∈S F (σ, U ).Robust schedules are desirable from a practical perspective because they hedge against adverse conditions of t...

Approximation Results for Makespan Minimization with Budgeted Uncertainty

“…In Section 2 we show that any c-approximation for the classical R||C max problem leads to a (c + 1)approximation for R|U Γ |C max , hence obtaining a 3-approximation algorithm for the latter problem, and a (2 + ǫ)-approximation for Q|U Γ |C max . We point out that this result improves the ad-hoc 3-approximation of [5] for P |U Γ |C max , while having a simpler proof. In Section 3, we show through a reduction from the Restricted Assignment Problem that there exists no (2 − ǫ)-approximation algorithm for R|U Γ |C max unless P = N P. In Section 4 we consider the P |U Γ |C max problem and present the first step our main result, namely a PTAS which is valid even when Γ is part of the input, i.e., not constant.…”

“…In this paper we improve the results of [5] for P |U Γ |C max and R|U Γ |C max . In Section 2 we show that any c-approximation for the classical R||C max problem leads to a (c + 1)approximation for R|U Γ |C max , hence obtaining a 3-approximation algorithm for the latter problem, and a (2 + ǫ)-approximation for Q|U Γ |C max .…”

“…Robust scheduling has been considered in the past, mostly for finite uncertainty sets without particular structure, see for instance [1,6,9,10,12]. More recently, [5,13,2] considered robust packing and scheduling with the budgeted uncertainty model U Γ from [4]. Specifically, [5] (with authors in common) provided a 3-approximation algorithm and a (1+ ǫ)-approximation (PTAS) for P |U Γ |C max but only for a constant Γ , as well as a randomized approximation algorithm for R|U Γ |C max having an average ratio of O(log(m)).…”

“…More recently, [5,13,2] considered robust packing and scheduling with the budgeted uncertainty model U Γ from [4]. Specifically, [5] (with authors in common) provided a 3-approximation algorithm and a (1+ ǫ)-approximation (PTAS) for P |U Γ |C max but only for a constant Γ , as well as a randomized approximation algorithm for R|U Γ |C max having an average ratio of O(log(m)). They also considered problem 1|U Γ | j w j C max , proving that the problem is N P-hard in the strong sense, and providing a polynomial-time algorithm when w j = 1 for j ∈ J.…”

“…In Section 3, we show through a reduction from the Restricted Assignment Problem that there exists no (2 − ǫ)-approximation algorithm for R|U Γ |C max unless P = N P. In Section 4 we consider the P |U Γ |C max problem and present the first step our main result, namely a PTAS which is valid even when Γ is part of the input, i.e., not constant. Having Γ in the input (and not constant) requires a totally different technique from the one used in [5]. The algorithm is turned into an EPTAS in Section 5, i.e., a PTAS where the dependency of ǫ is not in the exponent of the encoding length.…”