Mathematical Models, Heuristics and Algorithms for Efficient Analysis and Performance Evaluation of Job Shop Scheduling Systems Using Max-Plus Algebraic Techniques A dissertation presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Manjeet Singh December 2013 © 2013 Manjeet Singh. All Rights Reserved. 2 This dissertation titled Mathematical Models, Heuristics and Algorithms for Efficient Analysis and Performance Evaluation of Job Shop Scheduling Systems Using Max-Plus Algebraic Techniques by MANJEET SINGH has been approved for the Department of Mechanical and Systems Engineering the Russ College of Engineering and Technology by Robert P. Judd Professor of Industrial and Systems Engineering Dennis Irwin Dean, Russ College of Engineering and Technology 3 ABSTRACT SINGH, MANJEET, Ph.D., December 2013, Mechanical and Systems Engineering Mathematical Models, Heuristics and Algorithms for Efficient Analysis and Performance Evaluation of Job Shop Scheduling Systems Using Max-Plus Algebraic Techniques Director of Dissertation: Robert P. Judd (127 pp.) This dissertation develops efficient methods for calculating the makespan of a perturbed job shop. All iterative scheduling algorithms require their performance measure, usually the makespan, to be calculated during every iteration. Therefore, this work can enhance the efficiency of many existing scheduling heuristics, e.g. Tabu Search, Genetic Algorithms, Simulated Annealing etc. This increased speed provides two major benefits. The first is the capability of searching a larger solution space, and second is the capability to find a better solution due to the extra time. The following is a list of major highlights of this dissertation. Th e dissertation extends the hierarchical block diagram model formulation and composition that was originally proposed by Imaev[2]. An algorithm is developed that reduces the complexity of calculating the makespan of the perturbed schedule of job shop with no recirculation from O(MNlogMN) to O(N2), where M is the number of machines and N the number of parts. An efficient algorithm that calculates kleene star of a lower triangular matrix is ( ) which is presented. This algorithm has complexity of of the traditional approach. Finally, a novel pictorial methodology, called the SBA (Serial Block Addition), is developed to calculate the makespan of a perturbed job shop. A very efficient single perturbed machine scheduling algorithm, with complexity of O(N2), is derived using the 4 SBA method. The algorithm was tested on 10,000 randomly generated problems. The solutions provided by scheduling algorithm were 95.27% times, within a 3% deviation of the optimal solutions. 5 ACKNOWLEDGEMENTS I would like to express my deep gratitude to all the people who made this dissertation possible. First and foremost, a hearty thanks to my advisor Dr. Robert P. Judd. It was his vision and constant support which helped me through the continuous struggle of understanding the known and finding the unknown. Without his guidance and support, this dissertation wo uldn't have been possible. I will never forget one of his invaluable saying: “Most of the time there is a way to simplify a seemingly complex problem”. Additionally, I would like to thank Dr. Gursel Suer for all his support and guidance throughout my graduate studies at Ohio University. I would also like to thank all of the members of my dissertation committee, namely: Dr. Namkyu Park, Dr. Andy Snow and Dr. Ken Cutright. Their invaluable suggestions and constructive criticism helped me immensely during my research. I would also like to thank Tonya Seelhorst who always found a way to help me in my time of need, as well as my close friend, Tianjiao Chen, who was the source of inspiration which lead to this fruitful journey. I send a big hug and thanks to my sister, Indu, who I have missed having around immensely, for all the encouragement. I owe all my achievements to my parents, Mr. Sunder Singh and Mrs. Ram Rati, who have inspired me throughout my life with their hard work and perseverance. Finally, I would like to thank all my friends who still manage to love me despite all my idiosyncrasies. 6 TABLE OF CONTENTS Page Abstract…. ...........................................................................................................................3 Acknowledgements ..............................................................................................................5 List of Tables .......................................................................................................................8 List of Algorithms ..............................................................................................................10 List of Figures ......................................................................................................................9 Chapter 1: Introduction ......................................................................................................11 1.1 Brief Overview of the Problem .........................................................................11 1.2 Contributions .....................................................................................................12 Chapter 2: Literature Review .............................................................................................16 2.1 Max Plus Algebra in Scheduling .......................................................................16 2.2 Scheduling Techniques ......................................................................................21 2.2.1 Scheduling of Cyclic Systems ......................................................................21 2.2.2 Three Machine Scheduling ...........................................................................23 2.2.3 Enumerative Techniques...............................................................................24 2.2.4 Constructive Algorithms ...............................................................................25 2.2.5 Iterative Algorithms ......................................................................................25 2.3 Calculation of Makespan of a Job Shop ............................................................26 2.4 Missing Areas in Existing Research ..................................................................27 Chapter 3: Max Plus Algebra .............................................................................................28 Chapter 4: Block Diagram Modeling Approach ................................................................31 4.1 Applying the Block Diagram Approach to a Single Machine ...........................33 4.2 An Example Problem ........................................................................................36 Chapter 5: Modeling the Job Shop System ........................................................................39 5.1 Algorithm for Efficient Calculation of .........................................................42 5.2 Special Cases .....................................................................................................43 5.2.1 Generic Flow Shop .......................................................................................43 5.2.2 Flow Shop with all Jobs Flowing from G to F .............................................44 5.2.3 Recirculation of Jobs in G ............................................................................45 5.2.4 No Interaction between F and G...................................................................45 5.3 Example Problem ..............................................................................................46 Chapter 6: Bi-Part Modeling of a Job Shop Using Max Plus Algebra ..............................52 6.1 Determining the Makespan of a System ............................................................55 6.2 Calculation of , and .......................................................................55 6.3 Analysis of the Makespan equation of the System – A Special Case ...............60 Chapter 7: Modeling of a Job Shop Without Recirculation of Jobs .................................62 7.1 Makespan Equation for a Job Shop without Recirculation of Jobs ...................62 7.2 Calculation of F under Scheduling Perturbations for V Containing Single Machine .............................................................................................................62 7.3 Algorithm for Efficient Calculation of Makespan under Scheduling Perturbations for V ............................................................................................65 7.4 Example Problem ..............................................................................................67 7 Chapter 8: Modeling of Job Shop With Recirculation of Jobs ..........................................70 8.1 Example Problem ..............................................................................................73 8.2 Test for Feasibility of a Schedule ......................................................................74 8.3 Computation of the Star of a Lower Triangular Matrix ....................................76 8.4 Applying the Block Diagram Approach for Calculation of F for V Containing a Single Machine and Jobs going through Recirculation and Perturbation .........77 8.5 Algorithm to Calculate the Makespan of a Job Shop System with Recirculation and Reordering ..................................................................................................80