In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the
thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is
a model from convex optimization. This problem includes a large class of
linear programs. The algorithms generate approximately feasible solutions
within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and
O(M epsilon{-2} ln (M epsilon^{-1}))iterations,respectively,whereineachiterationablockproblemwhichdependsonthespecificapplicationhastobesolved.Bothalgorithms,appliedtolinearprograms,canresultincolumngenerationalgorithms.InChapter3,weimplementanalgorithmforthesoâcalledmaxâminâresourcesharingproblem.Thisisacertainconvexoptimizationproblemwhich,similartotheprobleminChapter1,includesalargeclassoflinearprograms.Theimplementation,whichisincludedintheappendix,isdoneinC++.WeusetheimplementationinthecontextofanAFPTASforStripPackinginordertoevaluatedynamicoptimizationofaparameterinthealgorithm,namelythesteplengthusedforinterpolation.Wecompareourchoicetothestaticsteplengthproposedintheanalysisofthealgorithmandconcludethatdynamicoptimizationofthesteplengthsignificantlyreducesthenumberofiterations.InChapter4,westudytwocloselyrelatedschedulingproblems,namelynonâpreemptiveschedulingwithfixedjobsandschedulingwithnonâavailabilityforsequentialjobsonmidenticalmachinesunderthemakespanobjective,wheremisconstant.Forthefirstproblem,whichdoesnotadmitanFPTASunlessP=NP,weobtainanewPTAS.Forthesecondproblem,weshowthatasuitablerestriction(namelythepermanentavailabilityofonemachine)isnecessarytoobtainaboundedapproximationratio.Forthisrestriction,whichdoesnotadmitanFPTASunlessP=NP,wepresentaPTAS;wealsodiscussthecomplexityofvariousspecialcases.Intotal,theresultsarebasicallybestpossible.InChapter5,wecontinuethestudiesfromChapter4wherenowthenumbermofmachinesispartoftheinput,whichmakestheproblemalgorithmicallyharder.Schedulingwithfixedjobsdoesnotadmitanapproximationratiobetterthan3/2,unlessP=NP;hereweobtainanapproximationratioof3/2+epsilonforanyepsilon>0.Forschedulingwithnonâavailability,werequireaconstantpercentageofthemachinestobepermanentlyavailable.Thisrestrictionalsodoesnotadmitanapproximationratiobetterthan3/2unlessP=NP;wealsoobtainanapproximationratioof3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is
refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible.
Finally, in Chapter 6, we conclude
with some remarks and open research problems
In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the
thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is
a model from convex optimization. This problem includes a large class of
linear programs. The algorithms generate approximately feasible solutions
within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and
O(M epsilon{-2} ln (M epsilon^{-1}))iterations,respectively,whereineachiterationablockproblemwhichdependsonthespecificapplicationhastobesolved.Bothalgorithms,appliedtolinearprograms,canresultincolumngenerationalgorithms.InChapter3,weimplementanalgorithmforthesoâcalledmaxâminâresourcesharingproblem.Thisisacertainconvexoptimizationproblemwhich,similartotheprobleminChapter1,includesalargeclassoflinearprograms.Theimplementation,whichisincludedintheappendix,isdoneinC++.WeusetheimplementationinthecontextofanAFPTASforStripPackinginordertoevaluatedynamicoptimizationofaparameterinthealgorithm,namelythesteplengthusedforinterpolation.Wecompareourchoicetothestaticsteplengthproposedintheanalysisofthealgorithmandconcludethatdynamicoptimizationofthesteplengthsignificantlyreducesthenumberofiterations.InChapter4,westudytwocloselyrelatedschedulingproblems,namelynonâpreemptiveschedulingwithfixedjobsandschedulingwithnonâavailabilityforsequentialjobsonmidenticalmachinesunderthemakespanobjective,wheremisconstant.Forthefirstproblem,whichdoesnotadmitanFPTASunlessP=NP,weobtainanewPTAS.Forthesecondproblem,weshowthatasuitablerestriction(namelythepermanentavailabilityofonemachine)isnecessarytoobtainaboundedapproximationratio.Forthisrestriction,whichdoesnotadmitanFPTASunlessP=NP,wepresentaPTAS;wealsodiscussthecomplexityofvariousspecialcases.Intotal,theresultsarebasicallybestpossible.InChapter5,wecontinuethestudiesfromChapter4wherenowthenumbermofmachinesispartoftheinput,whichmakestheproblemalgorithmicallyharder.Schedulingwithfixedjobsdoesnotadmitanapproximationratiobetterthan3/2,unlessP=NP;hereweobtainanapproximationratioof3/2+epsilonforanyepsilon>0.Forschedulingwithnonâavailability,werequireaconstantpercentageofthemachinestobepermanentlyavailable.Thisrestrictionalsodoesnotadmitanapproximationratiobetterthan3/2unlessP=NP;wealsoobtainanapproximationratioof3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is
refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible.
Finally, in Chapter 6, we conclude
with some remarks and open research problems
Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent Univ., 1991.Thesis (Ph. D.) -- Bilkent University, 1991.Includes bibliographical references.The primary concern of this study is to investigate the combinatorial aspects
of the single-stage identical parallel machine scheduling problem and to develop
a computationally feasible branch and bound algorithm for its exact solution.
Although there is a substantial amount of literature on this problem, most
of the work in this area is on the development and performance analysis of
approximation algorithms. The few optimizing algorithms proposed in the
literature have major drawbacks from the computer implementation point of
view. Even though the single-stage scheduling problem is known to be unary
A/âP-hard, there is still a need to develop a computationally feasible optimizing
algorithm that solves the problem in a reasonable time. Development of such
an algorithm is necessary for solving the multi-stage parallel machine scheduling
problems which are currently an almost untouched issue in the deterministic
scheduling theory.
In this study, a branch and bound algorithm for the single-stage identical parallel
machine scheduling problem is proposed. Promising results were obtained in
the empirical analysis of the performance of this algorithm. Furthermore,
the procedure that is developed to determine tight bounds at a node of the
enumeration tree, is an approximation algorithm that solves a special class
of identical parallel machine scheduling problems of practical interest. This
algorithm delivers a solution that is arbitrarily close to 4/3 times the optimum.
To our knowledge this is the best result obtained for this problem so far.Akyel, H CemalPh.D
Nine manuscripts were published in this Special Issue on âTheoretical and Computational Research in Various Scheduling Models, 2021â of the MDPI Mathematics journal, covering a wide range of topics connected to the theory and applications of various scheduling models and their extensions/generalizations. These topics include a road network maintenance project, cost reduction of the subcontracted resources, a variant of the relocation problem, a network of activities with generally distributed durations through a Markov chain, idea on how to improve the return loading rate problem by integrating the sub-tour reversal approach with the method of the theory of constraints, an extended solution method for optimizing the bi-objective no-idle permutation flowshop scheduling problem, the burn-in (B/I) procedure, the Pareto-scheduling problem with two competing agents, and three preemptive Pareto-scheduling problems with two competing agents, among others. We hope that the book will be of interest to those working in the area of various scheduling problems and provide a bridge to facilitate the interaction between researchers and practitioners in scheduling questions. Although discrete mathematics is a common method to solve scheduling problems, the further development of this method is limited due to the lack of general principles, which poses a major challenge in this research field
Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled âAdvances and Novel Approaches in Discrete Optimizationâ. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms
This dissertation comprises five papers, which have been published in scientific journals between 2019 and 2022. The papers consider logistic optimization problems from three different subjects with a focus on intra-logistics. All considered optimization problems have strong combinatorial aspects. To solve the considered problems, various solution approaches including different decomposition techniques are employed.
Paper 1 investigates the optimization of the layout and storage assignment in warehouses with U-shaped order picking zones. The paper considers two objectives, namely minimizing the order picker's walking distance and physical strain during order picking. To solve the problem, a semantic decomposition approach is proposed, which solves the problem in polynomial time. In a computational study, both considered objectives are found to be mostly complementary. Moreover, suggestions for advantageous layout designs and storage assignments are derived.
Paper 2 considers the problem of how to stow bins on tow trains in order to minimize the handling personnel's physical strain for loading and unloading. The problem is shown to be NP-hard and decomposed semantically. Utilising the decomposition, the problem is solved exactly with dynamic programming and heuristically with a greedy randomized adaptive search procedure. A consecutive computational study shows that both procedures perform well. Beyond that, it investigates the influence of the tow train wagons' design on the considered objective.
Paper 3 is concerned with the problem of scheduling jobs with time windows on unrelated parallel machines, which is a NP-hard optimization problem that has applications, i.a., in berth allocation and truck dock scheduling. The paper presents an exact logic-based Benders decomposition procedure and a heuristic solution approach based on a set partitioning formulation of the problem. Moreover, three distinct objectives, namely minimizing the makespan, the maximum flow time, and the maximum lateness are considered. Both procedures exhibit good performances in the concluding computational study.
Paper 4 addresses the problem of order picker routing in a U-shaped order picking zone with the objective of minimizing the covered walking distance. The problem is proven to be NP-hard. An exact logic-based Benders decomposition procedure as well as a heuristic dynamic programming approach are developed and shown to perform well in computational tests. Beyond that, the paper discusses different storage assignment policies and compares them in a numeric study.
Paper 5 studies scheduling electrically powered tow trains in in-plant production logistics. The problem is regarded as an Electric Vehicle Scheduling Problem, where tow trains must be assigned to timetabled service trips. Since the tow trains' range is limited, charging breaks need to be scheduled in-between trips, which require detours and time. The objective consists in minimizing the required fleet size. The problem is shown to be NP-hard. To solve the problem, Paper 5 proposes a branch-and-check approach that is applicable for various charging technologies, including battery swapping and plug-in charging with nonlinear charge increase. In a computational study, the approach's practical applicability is demonstrated. Moreover, influences of the batteries' maximum capacity and employed charging technology are investigated