CHAPTER 1 Basic Concepts on Planning and Scheduling Scheduling, FEUP/PRODEI /MIEIC 1 Planning and Scheduling: Processes of Decision Making regarding the selection and ordering of activities as well as the optimization of limited resources used for executing them in production environments (including services) Planning and Scheduling Processes are based in techniques and methods: mathematical heuristic Scheduling, FEUP/PRODEI /MIEIC 2 1
Planning and Scheduling: Are of great importance for large projects/works that encompass many tasks involving some kind of Restrictions (either mutual or not) Objective usually is to minimize the deadline of the last Task - Makespan Scheduling, FEUP/PRODEI /MIEIC 3 For the production lines (Job Shop) in the context of Flexible Manufacturing Systems (FMS) the goal is to Maximize the throughput (transfer rate; output relative to input; the amount passing through a system from input to output) FMS are production systems with a high degree of automatisation. As an example, we point the Automotive Industry Scheduling, FEUP/PRODEI /MIEIC 4 2
Planning and Control of Processes and projects Application Example : Construction, set up and test of a large computer system Tasks: hardware selection development/adaptation of software test and error debugging Recruiting and training of operators Objectives: Minimal Setup Time In accordance with Task ordering Scheduling, FEUP/PRODEI /MIEIC 5 Application Example: Integrated Circuits Production System Tasks: Silicon Wafers production Silicon Wafers test Cutting the Integrated Circuits Assembling the Integrated Circuits Quality Tests Objectives: To produce as much as possible of high quality ICs Scheduling, FEUP/PRODEI /MIEIC 6 3
Planning finds out, at a higher level of abstraction and aggregation, those restrictions to be followed in the next stages by Scheduling and Decision Making processes at a more detailed level Scheduling, FEUP/PRODEI /MIEIC 7 Example: planning and Scheduling for a supply chain. Tasks: material and goods are moved from one company to another (in a network of enterprises). Objectives: minimize total costs (production, transportation and inventory). paper mill is included in a network of enterprises starting from the one who supplies the wood, cellulose (pulp), until the final consumers of paper. Additional value is being added in each one of the chain steps. Scheduling, FEUP/PRODEI /MIEIC 8 4
Scheduling, FEUP/PRODEI /MIEIC 9 Anticipate/influencing Consumer Behaviours Maximizing Profits Scheduling, FEUP/PRODEI /MIEIC 10 5
Supply-Chaining Models xx Scheduling, FEUP/PRODEI /MIEIC 11 Characteristics of Activities (processing) Restrictions about: precedence eligible machines and tools workforce (teams, workers, workload ) handling of materials (robots, ) Restrictions about: costs setup time and sequence, maintenance, storage space, waiting times priorities (task interruption/abortion due to other events) transportation capabilities Scheduling, FEUP/PRODEI /MIEIC 12 6
Performance Measures and Objectives: Throughput: depending on critical paths (or machines) (bottlenecks) Makespan: Cmax = max (C1, C2,, Cn) where Ci is the deadline for executing task Ti Minimizing Makespan is equivalent to maximizing Throughput Scheduling, FEUP/PRODEI /MIEIC 13 Performance Measures and Objectives: Temporal Deviation (Lateness): Lj = Cj dj Where dj is the due time to execute Task j Maximum Deviation : max (L1, Ln) Tardiness: Objective Function Tj = max(cj-dj, 0) Weighted Tardiness: 1 to n Wj*Tj Scheduling, FEUP/PRODEI /MIEIC 14 7
Performance Measures and Objectives: Objectives usually combine time and resources utilization (costs) Minimizing total Costs, related with: makespan setup tardiness personal Try to find out the ideal time t 0 for executing the activities Scheduling, FEUP/PRODEI /MIEIC 15 Plan of Activities Representation : Precedences: 1-4 2-5 3-6, 7 4-6, 7 5-6 6-2 7-2 2-8 8 - Tasks on the temporal axis 8 Scheduling, FEUP/PRODEI /MIEIC 16 8
Projects and Processes Planning and Control Schedule the activities in a temporal scale Task i Processing Times Pi are known Assume resources independency Enough number of Machines in parallel N Tasks with precedence constraints Objective: minimizing makespan There are Tasks whose starting time may be postponed (slack job) And others that are seen as critical tasks The set of Tasks in the longest path between the starting node and end node, path called the critical path, includes Tasks that are Critical: Any delay will produce also a delay in the end of the project Scheduling, FEUP/PRODEI /MIEIC 17 Projects and Processes Planning and Control Advantages: Graphical vision (Net) of the project/ process Anticipates the needed time to conclude the project/process Makes explicit those critical activities to take care of in the Plan Stages: Specify individual activities Determine the sequence of those activities Generate the diagram as a network Estimate the needed time for each activity Identify the critical path Revise the diagram during the execution stage Scheduling, FEUP/PRODEI /MIEIC 18 9
forward procedure Starting the analysis at t=0 calculates the earliest possible time instant in which a task has to start Calculates the time instant at which last Task ends= makespan backward procedure Starting the analysis at time t= makespan calculates maximum time at which tasks may start. Finds out the Critical path Scheduling, FEUP/PRODEI /MIEIC 19 Notation: _pj task j processing time _S'j task j soonest possible starting time _C'j task j soonest possible ending time _S"j task j latest possible starting time _C"j task j latest possible ending time _C'j = S'j + pj _{all k j} tasks preceding task j _{j all k } tasks following task j Scheduling, FEUP/PRODEI /MIEIC 20 10
Forward Procedure : Step 1: t = 0 Do S j=0 and C j= pj. For all Tasks j without predecessors Step 2: Calculate for each Task j S j= max C k {all k j} C j= S j + pj Step 3: Optimal makespan is: C max= max(c 1,..,C n) Scheduling, FEUP/PRODEI /MIEIC 21 Backward Procedure : Step 1: t = Cmax Do C j= Cmax and S j= Cmax - pj. For all Tasks j without successors Step2: Calculate for each Task j C j = min S k (j all k ) (max P k ) S j = C j - pj Step 3: Verify that: min(s 1,..,S n)=0 Scheduling, FEUP/PRODEI /MIEIC 22 11
Comments: Forward Procedure calculates for each task the soonest possible starting time Backward Procedure calculates the latest possible starting time for each Task If both these time instants are equal, the task belongs to the Critical Path If they are different, the Task is considered a slack job ( relaxed task ) Scheduling, FEUP/PRODEI /MIEIC 23 Example: Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Pj 5 6 9 12 7 12 10 6 10 9 7 8 7 5 Precedence: Scheduling, FEUP/PRODEI /MIEIC 24 12
Forward Procedure: Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Pj 5 6 9 12 7 12 10 6 10 9 7 8 7 5 Cmax = 56 Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C j 5 11 14 23 21 26 33 32 36 42 43 51 50 56 Scheduling, FEUP/PRODEI /MIEIC 25 Backward Procedure: Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Pj 5 6 9 12 7 12 10 6 10 9 7 8 7 5 Cmax = 56 Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C j 5 12 14 24 30 26 34 36 36 43 43 51 51 56 Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C j 5 11 14 23 21 26 33 32 36 42 43 51 50 56 Scheduling, FEUP/PRODEI /MIEIC 26 13
Critical Path method (CPM) Critical Path : Cmax = 56 Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Scheduling, FEUP/PRODEI /MIEIC 27 There are four kinds of dependencies with respect to ordering terminal elements (in order of decreasing frequency of use): 1.Finish to start (FS) A FS B = B doesn't start before A is finished (Foundations dug) FS (Concrete poured) 2. Finish to finish (FF) A FF B = B doesn't finish before A is finished o (Last chapter written) FF (Entire book written) 3. Start to start (SS). A SS B = B doesn't start before A starts o (Project work started) SS (Project management activities started) 4. Start to finish (SF) A SF B = B doesn't finish before A starts Scheduling, FEUP/PRODEI /MIEIC 28 14
There are three kinds of dependencies with respect to the reason for the existence of dependency: 1. Causal (logical) It is impossible to edit a text before it is written It is illogical to pour concrete before you dig the foundations 2. Resource constraints It is logically possible to paint four walls in a room simultaneously but there is only one painter 3. Discretionary (preferential) I want to paint the living room before painting the dining room, although I could do it the other way round, too Traditional critical path-derived schedules are based only on causal (logical) dependencies Scheduling, FEUP/PRODEI /MIEIC 29 Program (or Project) Evaluation and Review Technique (PERT) a model for project management designed to analyze and represent the tasks involved in implementing a given project. It is commonly used in conjunction with the critical path method or CPM. Processing Times are non deterministic. Represented through stochastic variables Processing Times are averages M j and with a deviation (s j ) 2. Calculates makespan p ja =task j optimistic processing time (the minimum possible time required to accomplish a task) p jm = task j most likely processing time (the best estimate of the time required to accomplish a task) p jb = task j pessimistic processing time (the maximum possible time required to accomplish a task) M^j = task j expected processing time (the best estimate of the average time required to accomplish a task) Scheduling, FEUP/PRODEI /MIEIC 30 15
Program (or Project) Evaluation and Review Technique (PERT) Expected Makespan : Task j Average Processing Time (expected): M^j=(p ja +4p jm +p jb )/6 apply CPM with expected processing times Let Jcp be an average critical path Estimated expected Makespan: E^(Cmax)= j M^j (j belongs to the expected critical path) Scheduling, FEUP/PRODEI /MIEIC 31 Tasks processing Times. Restrictions same as previously considered Program (or Project) Evaluation and Review Technique (PERT) Calculation of averages and variances: If μ = E(X) is the expected time (average) for the random variable X, then the deviation is Var(X)= E ( (X-m) 2 ) i.e, the variance is the expected value of the squared difference between the variable's realization and the variable's mean. We may say in simple words that is the "Average of the squares of the distances between each time point and its average ". Thus it is the average of the squares of the deviations". The variance of a random variable "X" is Var(X) or simply σ 2. In this case σ 2 = (optimistic-pessimistic/6) 2 Scheduling, FEUP/PRODEI /MIEIC 32 16
Program (or Project) Evaluation and Review Technique (PERT) Eg. for the calculation of Var(Tj): Task T1: Var(T1)= [(4-6)/6] 2 =(1/3) 2 Task T2: Var(T2)= [(4-8)/6] 2 =(2/3) 2 Task T6: Var(T6)=[(12-12)/6] 2 = 0 Scheduling, FEUP/PRODEI /MIEIC 33 Program (or Project) Evaluation and Review Technique (PERT) Precedences were the same and the Critical path is: 1 3 6 9 11 12 14 Estimated Makespan : E^(Cmax)= m^j =56 (J belonging to Jcp) Makespan Estimated Variance :V^(Cmax)= s j 2 = 2.66 (J belonging to Jcp) Potential problems in using PERT: Sub-estimation of the project duration. Ignores those paths that are non critical It is probabilities-based Scheduling, FEUP/PRODEI /MIEIC 34 17