Elements of Sequencing and Scheduling by Kenneth R. Baker Introduction Chapter 1 Elements of Sequencing and Scheduling by Kenneth R. Baker Byung-Hyun Ha

Outline Schedule and scheduling Mathematical models Categorization and description Solution to scheduling problem Algorithms and time complexity

Introduction to Sequencing and Scheduling Schedule A tangible plan or document a bus schedule, a class schedule a. When things will happen; a plan for the timing of certain activities when will be the dinner served, when the laundry will be done b. In which sequence things will happen the North bus departs right after cleaning and maintenance are finished Scheduling Process of generating schedule Given tasks to be carried out, resources available to perform the tasks Determining the detailed timing of the tasks within resource capacity preparing a dinner, doing the laundry, problem in industry

Introduction to Sequencing and Scheduling Decision-making hierarchy Scheduling follows more basic decisions, some earlier dinner preparation  menu items, recipes, ... planning decisions in industry demand, design, technology for products A production planning and control hierarchy for pull system (next slide) Assuming planning decisions have been made already, and we are given the following information Tasks that are well defined and completely known Resources to perform these tasks are specified and available

Capacity/facility planning Sequencing & scheduling Marketing parameters Forecasting Production/process parameters Capacity/facility planning Workforce planning Labor policies Capacity plan Personnel plan Aggregate planning Aggregate plan Strategy WIP/quota setting Customer demand Master production schedule Demand management WIP position Sequencing & scheduling Tactics Control Real-time simulation Work schedule Work forecast Shop floor control Production tracking A production planning and control hierarchy for pull system (source: Hopp & Spearman, Factory Physics, 2000)

Introduction to Sequencing and Scheduling Defining scheduling problem Resource: type, amount Task: resource requirement, duration, available time, due date, technological constraints ( precedence restriction) Finding solution Formal problem-solving approaches are required because of complexity Formal models For understanding problem and finding solution Graphical, algebraic, and simulation models Gantt chart Visualizing problem, measuring performance, and comparing schedules Resource 1 1 2 4 3 Resource 2 2 1 4 3 Resource 3 3 2 1 4 time

Scheduling Theory Mathematical models Objective function Primary concerns in this course Development of useful models Leading to solution techniques and practical insights Interface between theory and practice Quantitative approach Capturing problem structure in mathematical form Beginning with a description of resources and tasks with translation of decision-making goals into explicit objective function Objective function Ideally, it should consist of all costs that depend on scheduling decisions Not practical (difficult to identify, isolate, and fix) Practical and prevalent goals Turnaround -- time required to complete a task Timeliness -- conformance of a task’s completion to a given deadline Throughput -- amount of work completed during a fixed period of time

Scheduling Theory Categorization of scheduling models By number of machines one or several By capacity of machine available in unit amount or in parallel By job availability static -- available jobs does not change over time, or dynamic -- new jobs appear over time Although dynamic models, which is less tractable than static models, would appear to be more important for practical application, static models often capture the essence of dynamic systems, and the analysis of static problems frequently uncovers valuable insights and sound heuristic principles that are useful in dynamic situation. By certainty deterministic or stochastic

Scheduling Theory Description of scheduling problem (source: Pinedo, 2008) Using triple  |  |   -- machine environment  -- processing characteristics and constraints  -- objective to be minimized Machine environment Single machine -- 1, identical machines in parallel -- Pm, flow shop -- Fm, job shop -- Jm, ... Processing characteristics and constraints Release dates -- rj, preemptions -- prmp, precedence constraints -- prec, sequence dependent setup times -- sjk, batch processing -- batch(b), ... pj = p, dj = d, ... Objective to be minimized Makespan -- Cmax, maximum lateness -- Lmax, total weighted completion time -- wjCj, weighted number of tardy jobs -- wjUj, ... Examples 1 | rj, prmp | wjCj, 1 | sjk | Cmax, Fm | pij = pj | wjCj, Jm || Cmax

Scheduling Theory Solution to scheduling problem Answering the following decisions Which resources will be allocated to perform each task? (allocation decision) When will each task be performed? (sequencing decision) by feasible resolution of the following common constraints Limits on capacity of machines Technological restriction on order in which some jobs can be performed ... Solution methodologies Mathematical programming models Combinatorial procedures Simulation techniques Heuristic solution approaches

Scheduling Theory Algorithms and time complexity Time complexity Computing effort required by a solution algorithm The number of computations required by an algorithm to solve a problem of “size n (amount of information needed to specify problem)” Size of sorting problem -- number of entries Size of travelling salesman problem (TSP) -- number of cities, roughly Described by order-of-magnitude notation, O() A function f(n) is O(g(n)) whenever there exists a constant c such that |f(n)|  c|g(n)| for all values of n  0 Insertion sort -- O(n2), quick sort -- O(nlogn) All enumeration algorithm: knapsack problem -- O(2n), TSP -- O(n!) http://en.wikipedia.org/wiki/Computational_complexity_theory#Big_O_notation Polynomial time algorithm An algorithm with complexity such as O(1), O(n2), O(n8), ... Exponential time algorithm An algorithm with complexity such as O(2n), O(3n), ...

 105 sec.  1 day, 41017 sec.  age of the universe Scheduling Theory Algorithms and time complexity (cont’d) Running time example Comparing algorithms with complexities of n, nlogn, n2, n8, 2n, 3n Assuming that our computer executes 106 computations per second Assuming that our computer has become 103 times faster than above  105 sec.  1 day, 41017 sec.  age of the universe

With computer 100 times faster With computer 1000 times faster Scheduling Theory Algorithms and time complexity (cont’d) Running time example (cont’d) Size of largest problem instance solvable in 1 hour (source: Garey and Johnson, 1969) Time complexity With present computer With computer 100 times faster With computer 1000 times faster n n2 n3 n5 2n 3n N1 N2 N3 N4 N5 N6 100 N1 10 N2 4.64 N3 2.5 N4 N5 + 6.64 N6 + 4.19 1000 N1 31.6 N2 10 N3 3.98 N4 N5 + 9.97 N6 + 6.29

Scheduling Theory Algorithms and time complexity (cont’d) NP-complete problems A class of problems which includes many well-known and difficult combinatorial problems Knapsack problem, TSP, graph coloring problem, ... All the problems are equivalent in the sense that if one of them can be solved by polynomial algorithm, then so can the others No one has found polynomial algorithm, nor proved no algorithm exists Garey and Johnson, 1969 http://en.wikipedia.org/wiki/Np-complete Proof of NP-completeness of a problem By reducing a well-known NP-complete problem to the problem Application of NP-complete theory If we are faced with the need to solve large versions of an NP-complete problem, we might be better off to use a so-called heuristic solution procedure, rather than pursuing optimal solution