chapter 1 basic concepts on planning and scheduling

Eugénio Oliveira FEUP/2015

MPE/ProDEI/MIEIC Eugénio Oliveira Planning and Scheduling, FEUP/PRODEI /MIEIC 1 Planning and Scheduling, FEUP/PRODEI /MIEIC 1

Eugénio Oliveira

CHAPTER 1

Basic Concepts on Planning and Scheduling

Methods for Planning and Scheduling




Planning and Scheduling: Processes of “Decision Making” regarding the 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



Planning and Scheduling: Are of great importance for large projects/works that encompass many tasks and goals involving some kind of restrictions (either mutual or not)

The Objective usually is to minimize the deadline of the last Task - “Makespan”




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 automation and flexibility. As an example, we point out the Automotive Industry


Job Shop: small and medium size job production by manufacturing systems

Productivity: production as a function of production factors



Application Example : Specify, set up and test of a large computer system Tasks: requirements identification hardware selection development/adaptation of software test and error debugging Recruiting and training of operators Objectives: Minimal Setup Time In accordance with Task ordering restrictions

Planning and Control of Processes and projects




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




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




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, inventory, …).

Ex.: paper mill is included in a network of enterprises starting from the one who supplies the wood, cellulose (pulp), paper finishing, storing, until the end-consumers of the paper (in a stationery). Additional value is being added in each one of the chain steps.




Métodos de Planeamento e Escalonamento

xx

Supply-Chain Models



Anticipates/influencing Consumer Behaviours Maximizing Profits




Characteristics of Activities (processing)

Restrictions about: precedence eligible machines and tools workforce (teams, workers, workload…) handling of materials (robots,…)

Restrictions about: costs setup time, maintenance,… storage space, waiting times priorities (task interruption/abortion due to other events) transportation capabilities




Performance Measures and Objectives:

Minimizing “Makespan” is equivalent (helps) to maximizing “Throughput”

Throughput: depending on critical paths (or machines) (bottlenecks)

Makespan: Cmax = max (C1, C2,…, Cn) where Ci is the deadline for executing task Ti





Temporal Deviation (Lateness): Lj = Cj – dj Where dj is the due time to execute Task j. Lateness may be negative. Maximum Deviation : max (L1,…Ln)

Tardiness: Objective Function Tj = max(Cj-dj, 0) similar to Lateness but only positive.

Weighted Tardiness: 1 to n Wj*Tj


A scheduliong performance measure may be trying to minimizing total Tardiness



Objectives usually combine time and resources utilization (costs) Minimizing total Costs, related with: makespan setup tardiness personal

Try to find out the ideal time t0 for executing the activities





Plan of Activities Representation :

Precedences: 1 - 4 2 - 5 3 - 6, 7 4 - 6, 7 5 - 6 6 -2 7 -2 2-8 8 -


8

Tasks on the temporal axis

Network Diagram



Critical Path Method (CPM)

Schedule the activities in a temporal scale For all i:Task i Processing Times Pi are known Assumes resources independency Enough number of Machines in parallel (availability) 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, is called the “critical path”, includes Tasks that are Critical: Any delay will produce also a delay in the end of the project

Projects and Processes Planning and Control





Advantages of CPM: • 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

Projects and Processes Planning and Control

Stages of CPM : • 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





forward procedure Starting the analysis at t=0 calculates the earliest possible time instant in which each 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 each task can start. Finds out the Critical path





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


C’’

S’

Slack

Pj C’

S’’

Name of Task j

Task description in the diagrams




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 (other Tasks) S’j= max C’k {all kj} C’j= S’j + pj

Step 3: Optimal makespan is: C’max= max(C´1,..,C´n)





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 (other Tasks j) C’’j = min S’’k (j all k ) S’’j = C’’j - pj

Step 3: Verify that: min(S’’1,..,S’’n)=0





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”)

Comments: Forward Procedure calculates for each task the soonest possible starting time Backward Procedure calculates the latest possible starting time for each Task





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

Precedences:


This Network can be replaced by the Gantt chart




Forward Procedure:

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

Cmax = 56

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





Backward Procedure:

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

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


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

Cmax = 56

36



Critical Path method (CPM)

Critical Path :

Cmax = 56

Tasks 1 2 3 4 5 6 7 8 9 10 11 12 13 14





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)

There are four kinds of dependencies with respect to ordering terminal elements (in order of decreasing frequency of use):

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

http://en.wikipedia.org/wiki/File:Dependency-FS.png

http://en.wikipedia.org/wiki/File:Dependency-FF.png

http://en.wikipedia.org/wiki/File:Dependency-SS.png

http://en.wikipedia.org/wiki/File:Dependency-SF.png




There are three kinds of reasons with respect to the existence of dependencies:

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



Program (or Project) Evaluation and Review Technique (PERT)

Processing Times are non deterministic. Represented through stochastic variables Processing Times are averages Mj and with a variance (sj)

2 (and deviation) . PERT Calculates makespan pja =task j optimistic processing time (the minimum possible time required to accomplish a task)

pjm = task j most likely processing time (the best estimate of the time required to accomplish a task)

pjb = 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)


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.



Expected Makespan : Task j Average Processing Time (expected): M^

j=(pja+4pjm+pjb)/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)





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 real 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 σ2. In this case σ2= (optimistic-pessimistic/6)2



Calculation of averages and variances:

Tasks’ processing Times. Restrictions same as previously considered

Precedences:



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





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)= sj 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



chapter 1 basic concepts on planning and scheduling

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