Scheduling of inventory releasing jobs to minimize a regular objective function of delivery times

In this note we provide new complexity and algorithmic results for scheduling inventory releasing jobs, a new class of single machine scheduling problems proposed recently by Boysen et al. We focus on tardiness related criteria, while known results are concerned with inventory levels between fixed delivery points. Our interest is motivated by the fact that deciding whether a feasible schedule exists is NP-hard in the strong sense, provided that all delivery deadlines are fixed, and there are no restrictions on the amount of products released by the jobs, nor on the job processing times. We will establish NP-hardness results, or provide polynomial or pseudo-polynomial time algorithms for various special cases, and describe a fully polynomial approximation scheme for one of the variants with the maximum tardiness criterion. © 2012 Springer Science+Business Media New York

Fixed job scheduling: a literature survey and a solution proposition through meta heuristic methods

Fixed Job Scheduling (FJS) is defined as the arrangement of the works, which have fixed ready times and deadlines in a shift with particular number of machines. In fixed job scheduling problems, the objective is to select a set of jobs for processing so as to maximize the total weight. This problem is known to be NP-hard.  The Fixed job scheduling problem has two variants based on objective functions (Eliiyi, Azizoğlu, 2006). The first variant is the Operational Fixed Job Scheduling (OFS), where each job j has a weight wj that represents its value or relative importance, and the concern is maximizing the total weight of the processed jobs with a given number of processors. The second variant is the tactical fixed job scheduling problem, which considers the minimization of the total cost or the number of the machines needed to process all jobs (Eliiyi, Azizoğlu, 2006). FJS has been studied extensively in recent years. Arkin and Silvenberg (1987) analyzed the jobs scheduling problems with fixed start and end times. They proposed an algorithm which maximizes the value of jobs completed by k identical machines. They also showed that the problem is NP-Complete. Kolen and Kron (1992) investigated FJS problem which appears in the aircraft maintenance process at an airport. They show that the polynomially solvable cases of these problems can be solved by a combination of linear programming and network flow algorithms. Dondeti and Emmons (1992) study fixed job scheduling problem that involves two types of processors but three types of jobs. They present a polynomial algorithm for finding the minimal cost combination of the two types of processors required to complete all jobs. Fischetti et al. (1992) introduce several polynomial-time approximation algorithms for fixed job schedule problems. Some of the algorithms they present make use of a simple procedure for assigning to processors, in a greedy way. A later study by Kolen and Kroon (1994) addresses the fixed job schedule problem which appears in the aircraft maintenance process at an airport. They present an analysis of the problem of finding the minimum total number of engineers required for carrying out all jobs. In their study the engineers are addressed as machines. Another study of Kroon et al. (1995) is exact and approximation algorithms for the operational fixed interval scheduling problem. They discuss the occurrence of the fixed interval scheduling problems in practice and develop exact and approximation algorithms for solving OFS problem. Bouzina and Emmons (1996) present a polynomial solution to several interval scheduling problems. The objective of their algorithm is maximizes the number of processed jobs.  A later study of Kroon et al.(1997) is exact and approximation algorithms for the tactical fixed interval scheduling problem. They present exact and approximation algorithms for solving the tactical fixed interval scheduling problem.  In the first study on interval scheduling problems to solve by a Meta heuristic method has been proposed by Santos and Zhong (2001).  They developed a Genetic algorithm (GA) and reinforcement learning for the tactical fixed interval scheduling problem.. A graph based heuristic is described to solve the operational fixed job scheduling problem by Garcia et al. (2005). They compared their solution with other heuristic from literature.  Eliiyi and Azizoğlu (2006) propose a branch and bound algorithm for solving operational fixed job scheduling problem. Despite its practical importance and broad range of usage area from the maintenance process of planes at the airports to the car repair /rental services, there are very few studies in literature on the subject of fixed job scheduling problem. In this study, a literature survey on fixed job scheduling problems has been done for the last twenty years and also two meta heuristic methods such as Genetic algorithm(GA) and Simulated annealing(SA) are proposed for solving the operational fixed job shop scheduling problems.  To show the performance of Genetic algorithm and simulated annealing, an example is presented in the study. A random test problem for the operational fixed job scheduling on identical parallel machine is solved with Bouzina and Emmons (1996) algorithm, and proposed GA and SA algorithms. The computational results indicate that the proposed metaheuristic methods are effective for operational fixed job scheduling problems. Keywords: Fixed job scheduling, Genetic algorithm, Simulated annealing.  Sabit iş çizelgeleme; sabit bir başlangıç ve bitiş zamanında, tamamlanması gereken işlerin ve belirli sayıda makinelerin bulunduğu bir vardiyada, işlerin çizelgelenmesi olarak tanımlanmaktadır. Sabit iş çizelgeleme, amaç fonksiyonuna göre iki temel alt problemden oluşmaktadır. Bunlar; sabit geliş ve teslim zamanına sahip işlerin, özdeş paralel makinelerde işlem görmek üzere, her işin wj ile ağırlıklandırılması sonucunda, toplam kâr maksimizasyonu amaçlı olan, Operasyonel sabit iş çizelgeleme ve sabit geliş ve teslim zamanlı işlerin ck sabit maliyetli paralel özdeş makineler üzerinde çizelgelenmesi ile maliyet minimizasyonu amaçlı olan, taktiksel sabit iş çizelgeleme problemleridir (Eliiyi, Azizoğlu, 2006). Havalimanında uçak bakım sürecinden, araç tamir/kiralama sistemlerine kadar geniş kullanım alanı bulunan sabit iş çizelgeleme probleminin, pratik önemine rağmen, literatürde çok az çalışma yapıldığı bilinmektedir. Bu araştırmada, sabit iş çizelgeleme problemi üzerine son yirmi yılda literatürde yapılan çalışmalar incelenmiştir. Ayrıca meta sezgisel yöntemlerde olan, Genetik Algoritma (GA) ile Tavlama Benzetiminin (TB) Operasyonel sabit iş çizelgeleme problemlerinin çözümünde kullanımı üzerine öneride bulunulmuştur. Bouzina ve Emmons (1996) tarafından geliştirilen algoritma (klasik yöntem) ile meta sezgisel yöntemlerden olan Genetik algoritmalar ve Tavlama benzetiminin, Operasyonel sabit iş çizelgeleme problemleri üzerindeki çözüm performansı, bir örnek üzerinde karşılaştırılmıştır. Araştırmada, meta sezgisel yöntemlerin daha iyi sonuçlar verebileceği belirlenmiştir. Anahtar Kelimeler: Sabit iş çizelgeleme, genetik algoritma, tavlama benzetimi.&nbsp

Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page

Stochastic scheduling on unrelated machines

Two important characteristics encountered in many real-world scheduling problems are heterogeneous machines/processors and a certain degree of uncertainty about the actual sizes of jobs. The first characteristic entails machine dependent processing times of jobs and is captured by the classical unrelated machine scheduling model.The second characteristic is adequately addressed by stochastic processing times of jobs as they are studied in classical stochastic scheduling models. While there is an extensive but separate literature for the two scheduling models, we study for the first time a combined model that takes both characteristics into account simultaneously. Here, the processing time of job $j$ on machine $i$ is governed by random variable $P_{ij}$, and its actual realization becomes known only upon job completion. With $w_j$ being the given weight of job $j$, we study the classical objective to minimize the expected total weighted completion time $E[\sum_j w_jC_j]$, where $C_j$ is the completion time of job $j$. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee $(3+\Delta)/2+\epsilon$. Here, $\epsilon>0$ is arbitrarily small, and $\Delta$ is an upper bound on the squared coefficient of variation of the processing times. We show that the dependence of the performance guarantee on $\Delta$ is tight, as we obtain a $\Delta/2$ lower bound for the type of policies that we use. When jobs also have individual release dates $r_{ij}$, our bound is $(2+\Delta)+\epsilon$. Via $\Delta=0$, currently best known bounds for deterministic scheduling are contained as a special case

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Greed Works -- Online Algorithms For Unrelated Machine Stochastic Scheduling

This paper establishes performance guarantees for online algorithms that schedule stochastic, nonpreemptive jobs on unrelated machines to minimize the expected total weighted completion time. Prior work on unrelated machine scheduling with stochastic jobs was restricted to the offline case, and required linear or convex programming relaxations for the assignment of jobs to machines. The algorithms introduced in this paper are purely combinatorial. The performance bounds are of the same order of magnitude as those of earlier work, and depend linearly on an upper bound on the squared coefficient of variation of the jobs' processing times. Specifically for deterministic processing times, without and with release times, the competitive ratios are 4 and 7.216, respectively. As to the technical contribution, the paper shows how dual fitting techniques can be used for stochastic and nonpreemptive scheduling problems.Comment: Preliminary version appeared in IPCO 201