Scheduling

Scheduling (Carlier & Chrtienne, 1988) Scheduling is to forecast the processing of awork by assigning resources to tasks and fixing their start times

The different components of a scheduling problem are the tasks, the potentialconstraints, the resources and the objective function The tasks must be programmed to optimise a specific objective Of course, often it will be more realistic in practice to consider several criteria

(Pinedo, 1995) Scheduling concerns the allocation of limited resources totasks over time. It is a decision-making process that has a goal the optimization of one or moreobjectives

(Baker, 1974) Scheduling is the allocation of resources over time toperform a collection of tasks Scheduling is a decision-making function: it is the process of determining aschedule Scheduling is a body of theory: it is a collection of principles, models,techniques, and logical conclusions that provide insight into the schedulingfunction

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The scheduling function The scheduling function is determining answers to the fundamentalmanagerial decisions questions:

What product or service is to be provided? On what scale will it be provided? What resources will be made available?

The scheduling function presumes that answers to these questionsis already exist The function scheduling becomes relevant in a situation where thenature of the tasks to be scheduled has been described and theconfiguration of the resources available has been determined

The scheduling and planning functions may not be completelyindependent Systems approach: the steps by which scheduling decisions arereached

Formulation, analysis, synthesis, evaluation

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Systems approach: 4 primary stages Formulation

The process of identifying the problem and determining the criteria that should guide decisionmaking Subtle and complicated activity Good decision can be seldom be expected without a clear definition of the problem at handand an explicit recognition of objective

Analysis The detailed process of examining the elements of problem and their interrelationships Aimed at identifying the decision variables and at specifying the relationships among themand the constraints they must obey

Synthesis The process of building alternative solutions to the problem The role is to characterize the feasible options that are available

Evaluation The process of comparing al feasible alternatives and selecting a desirable course of actionbased on criteria that were developed at the outset

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Gantt chart The simplest and most

widely used models Graphical

representation ofscheduling relationships

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Scheduling theory Scheduling theory is concerned primarily with mathematical models thatrelate to the scheduling function

The development of useful models and techniques has been the continuinginterfaces between theory and practice Quantitative approach

A translation of decision making goals into an explicit objective and decisionmaking restrictions into explicit constraints Decision Making Goals

Efficient utilization of resources Rapid response to demands Close conformance to prescribed deadlines

Feasibility constraints There are limits on the capacity of available resources There are technological restrictions on the order in which tasks can be performed

Solution: feasible resolution of the constraints Which resources will be allocated to perform each task? When will each task be performed?

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Scheduling theory The essence of scheduling problems:

Allocation decisions Sequencing decisions

The vital elements in scheduling models: Resources Tasks

The theory of scheduling includes a varietytechniques that are useful in solvingscheduling problem

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Scheduling problems withoutassignment

The problem is to find a processing start time for each operation. Several types of arrangement are traditionally encountered:

single machine: Only a single machine is available for the processing of jobs. It concerns a basic shop or one in which asingle machine poses a real scheduling problem. Besides, resolution of more complex problems isoften achieved by the study of single machine problems. We can find an area of direct application incomputing, if we think of the machine as the single processor of the computer. The jobs to beprocessed are necessarily mono-operation.

flowshop (F): several machines are available in the shop. The characteristic of this type of shop is that the jobsprocessed in it use machines in the same order: they all have the same processing routing. In apermutation flowshop we find in addition that each machine has the same sequence of jobs: theycannot overtake each other.

jobshop (J): several machines are available in the shop. Each job has a route of its own, i.e. it uses the resources inits own order.

openshop (O): several machines are available in the shop. The jobs do not have fixed routings. They can, therefore,use the machines in any order.

mixed shop (X): several machines are available in the shop. Some jobs have their own routing and others do not.

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Scheduling and assignment problemswith stages

The machines are grouped in well defined stages and a machine belongs to one stage only. In allcases the machines of a stage are capable of performing the same operations. To carry out oneoperation it is necessary to choose one among the available machines and, therefore, the problemis twofold, assigning one machine to each operation and sequencing the operations on themachines. At each stage we can differentiate between the following configurations:

the machines are identical (P): an operation has the same processing time on all the machines. the machines are uniform (Q): the processing time of an operation Oi,j on the machine Mk is equal to pi,j,k =qi,j/vk where qi,j is for example a number of components in the operation Oi,j to be processed, and vk is thenumber of components which the machine Mk can process per unit of time. the machines are unrelated or even independent (R): the processing time of the operation Oi,j on themachine Mk is equal to Pi,j,k, and is a data of the problem. Of course, just as the assignment of Oi,j isunknown, so is its processing time.

Globally, "traditional" scheduling and assignment problems correspond to the followingconfiguration: parallel machines (P/Q/R): there is only one stage and the jobs are mono-operation. hybrid flowshop (HF): all the jobs have the same production routing, and therefore use the stages in thesame order. general jobshop (GJ): each job has a route of its own. general openshop (GO): the jobs do not have a fixed routing.

It is easily possible to generalise these problems by supposing that each operation can only use itsown subset of the resources of the performing stage.

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General scheduling and assignmentproblems

This is the most general case where we suppose that eachoperation has its own set of machines on which it can be processed.No assumption is made on these sets of resources. We candifferentiate several cases: the jobs are mono-operations, and we are confronted by a problem ofparallel machines with general assignment. We find these problems inthe literature ([Brucker, 2004]) under the name "Multi PurposeMachines Scheduling Problems" (P/Q/R MPM SP). the jobs follow a processing order. It is difiicult in this case todistinguish between flowshop and jobshop since the groups ofmachines used by these jobs are not comparable. This is what is calledshops with general assignment problems (" General Shop MPM SP"). the jobs do not follow a fixed routing. This is the case in openshopwith general assignment problems ("Openshop MPM SP").

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constraints pmtn indicates that preemption is authorised.

Here it is possible to forsee interuption of an operation so that, possibly it can be taken upnext by another resource. split indicates that splitting is authorised.

Here it is possible to forsee splitting of the operation into lots, which can be performed on oneor several machines, possibly simultaneously. prec indicates that the operations are connected by precedence constraints.

This heading gives different particular cases according to the nature of the constraints: prec todescribe the most general case, tree, in-tree, outtree, chains and sp-graph (for series-parallelgraph ; see [Pinedo, 1995] or [Brucker, 2004]) to denote particular cases. batch indicates that the operations are grouped in batches.

Two types of batch constraints are differentiated in the literature: the first called sometimes s-batch concerns serial batches where the operations constituting a batch are processed insequence and the second of type p-batch concerns parallel batches where the operationsconstituting a batch are processed in parallel on a cumulative resource. In both cases, thecompletion time of an operation is equal to the completion time of the batch. In the first case.the duration of the batch is equal to the sum of the processing times of the operations whichconstitute it, whereas in the second case its duration is equal to the longest processing time ofthe operations in the batch.

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constraints no-wait indicates that the operations which constitute each job follow each otherwithout any waiting. prmu (permutation) indicates that the operations occur on the machines in thesame order. In other words, they cannot overtake themselves (this is true solely forflowshop problems). di = d indicates that all the due dates are identical. Likewise di d for thedeadlines. pi = p indicates that the processing times are all identical. We often encounter thisconstraint with p = 1. Snsd and Rnsd indicate that the setup and removal times on the resources beforeand after each processing, respectively, must be taken into account. Thesepreparation times are independent of the sequence of operations. Ssd and Rsd indicate that the setup and removal times on the resources before andafter each processing, respectively, must be taken into account. These preparationtimes are dependent of the sequence of operations.

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constraints ai1,i2 indicates that a minimum time lag must be respected between thejobs Jii and Ji2, if the jobs are mono-operation. Otherwise, we use ai1,j1,i2,j2to indicate a minimum time lag which must be respected between theoperation Oi1,j1 and the operation Oi2,j2. If this value is positive, we model,for example, a drying time between two successive operations, or else atransport time. In the latter case the resource is available to process thefollowing operation during the transport. If this value is negative, itindicates that it is possible to carry out an overlap, i.e. to start anoperation before its precedent in the routing is completely finished. Ofcourse, this is possible when a job is composed of lots of items and it isnot necessary to wait for the end of a lot on a machine to start theoperation on the following machine. blcg (balancing) is a constraint peculiar to parallel machine shops,translating the fact that the machines must complete processing of jobswhich are assigned to them at the same time. This constraint may beimposed when it is necessary to change the type of manufacture on all themachines simultaneously.

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constraints block (blocking) is a constraint indicating that the shop has a limitedstock area between the machines. Then we note bk the stockcapacity between machine Mk and machine Mk+1. recrc (recirculation) is a constraint which indicates that a job may beprocessed several times on the same machine. unavailj translates the case where all the resources are notavailable all the time, but only during well defined periods. It is amatter of timetables translating periods of opening/closing of thefactory, periods of planned maintenance, of holidays, etc. Two typesof operations can be associated with this problem: interruption andresumption of an operation as soon as possible (unavailj-resumable)or else the operation is not started if it is going to be interrupted(unavailj-nonresumable). In the latter case we can have problems ofunfeasibility.

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routing and precedence constraints a routing is a document which precisely describes the set ofoperations necessary for ending up with a final product:machine, processing time, tools, particular conditions, etc.This routing contains, of course, the order in which theoperations must be processed, possibly with the help of aprecedence graph (in the case of non identical routings).Two successive operations in a routing indicate a flow ofmaterial between machines or a set of machines. precedence relations between operations indicate simplythat the start of an operation is conditioned by the end ofall the preceding operations. No notion of flow is attached,a priori, to this constraint and it may simply be a matter ofsevere technological constraints. Two operations linked bya precedence relation may correspond to two distinct jobs.

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Types of Decision Making Goals Efficient utilization of resources Rapid response to demands Close conformance to prescribed deadlines

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Optimality criteria Minimisation of a maximum function:"minimax" criteria

Minimisation of a sum function: "minisum"criteria

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Minimisation of a maximum function:"minimax" criteria

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Minimisation of a sum function:"minisum" criteria

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Typology of scheduling problems

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Typology of scheduling problems

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Notation of problems Field refers to the typology and describes thestructure of the problem. It breaks down into two fields: = 12. The values of 1 and 2 refer to the machinesenvironment of the problem and possibly to thenumber of available machines.

Field contains the explicit constraints of theproblem.

Field contains the criterion/criteria to beoptimised.

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Notation of Data

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Notation of Variables

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