CZECH TECHNICAL UNIVERSITY IN PRAGUE FACULTY OF ELECTRICAL ENGINEERING DIPLOMA THESIS. Scheduling and Visualization of Manufacturing Processes

Transcription

1 CZECH TECHNICAL UNIVERSITY IN PRAGUE FACULTY OF ELECTRICAL ENGINEERING DIPLOMA THESIS Schedulng and Vsualzaton of Manufacturng Processes Prague, 2008 Author: Roman Čapek

2 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek

3 Dploma Thess CTU n Prague, 2008

4 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek v

5 Dploma Thess CTU n Prague, 2008 v

6 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek v

7 Dploma Thess CTU n Prague, 2008 Thanks I would lke to thank everbody who helped me ether drectly or ndrectly durng the work on ths thess. Especally to Ing. Přemysl Šůcha Ph.D., my supervsor, who has always gven me advce wllngly, sometmes even over the frame of hs dutes, and helped me wth the work flow and the fnalzaton of the thess. Furthemore, I would lke to thank to Doc. Dr. Ing. Zdeněk Hanzálek for hs expert advces and also to my parents who support me durng my entre study. v

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9 Dploma Thess CTU n Prague, 2008 Annotaton Ths work nvolves results acheved n two parts of the schedulng area. A research n the schedulng wth more alternatve process plans s descrbed n the frst place. We suggest to represent ths problem usng the Petr nets formalsm and we offer data structures and methods sutable for ths problem. For ths data representaton, an algorthm based on Integer Lnear Programmng (ILP) s proposed and tested on randomly-generated data. Second part of ths thess s dedcated to descrpton of utlzaton of the smulaton and vsualzaton n schedulng. Implementaton of VISIS (VIsualzaton and SImulaton n Schedulng), a tool for the smulaton and vsualzaton, s descrbed and case studes and examples are presented. Anotace Tato dplomová práce je věnována popsu výsledků dosažených ve dvou částech teore rozvrhování. Prvním řešeným problémem je rozvrhování s více alternatvním výrobním plány. V prác je zahrnut výzkum navazující na jž exstující práce v oblast alternatvního rozvrhování a pro zadání struktury problému je využta notace Petrho sítí. Pro takto popsaný problém je navržen algortmus založený na celočíselném lneárním programování. Navržené řešení je otestováno na náhodně generovaných nstancích problému s alternatvním výrobním plány. Druhá část této práce je potom věnována popsu využtí vzualzace a smulace v oblast rozvrhování. Práce rovněž obsahuje pops mplementace nového nástroje pro vzualzac a smulac nazvaného VISIS (VIsualzaton and SImulaton n Schedulng) a několk ukázkových příkladů vytvořených pomocí tohoto nástroje. x

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11 Dploma Thess CTU n Prague, 2008 Content Annotaton...x Content...x Lst of Fgures...xv 1. Introducton The Vsualzaton n Schedulng Schedulng and the Smulaton Optonal Schedulng Motvaton Contrbuton VISIS Applcaton Related Works for the Vsualzaton and Smulaton Algorthm for the Optonal Schedulng Problem Related Works for the Optonal Schedulng Problem Paper Organzaton Problem Descrpton Schedulng Theory Defnton of the Fundamental Schedulng Objects Examples of Schedulng Problems Gantt Chart The Host Schedulng Problem Optonal Schedulng Problem Statement Optonal Schedulng Problem Representaton Modfed Temporal Network - XOR Graph Petr Nets n Optonal Schedulng Petr Nets Bascs Converson of XOR Graph to Petr Net Algorthm for the Optonal Schedulng Problem Defnton of Parameters and Varables Integer Lnear Programmng Formulaton One Processor Infnte Amount of Identcal Parallel Processors Dedcated Processors Utlzaton of the Smulaton and Vsualzaton x

12 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek 5.1. Applcaton of the Smulaton Applcaton of the Vsualzaton Implementaton of VISIS Implemented Functons Smulnk Model Descrpton Task Representaton Vrtual Realty Toolbox Case Studes Performance Measures for the Optonal Schedulng Problem Smulaton wth VISIS Vsualzaton wth VISIS Profler Results Conclusons...55 References...57 A. User Manual...I A.1. Smulaton by Smpler Substtuton of TrueTme Lbrary... I A.2. Vsualzaton wth User-defned Vrtual Realty... III A.3. Defnton of Commands for Tasks...V x

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14 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Lst of Fgures ~ Fgure r, d Cmax problem Fgure Precedence constrants ~ Fgure Scheduled problem 1 r, d Cmax Fgure Scheduled problem P prec Lmax Fgure Host schedulng Gantt chart Fgure Host schedulng specal chart Fgure Alternatve process plans Fgure XOR graph Fgure Example of Petr net Fgure Choce representaton n PN Fgure Parallel process plans n PN Fgure Incorrect PN Fgure PN for the optonal schedulng problem Fgure Precedence constrants for DSVF flter Fgure Tasks executon Fgure Result of the smulaton Fgure MPC control Fgure Host schedulng problem Fgure Example of process flows Fgure S-Functon dalog box xv

15 Dploma Thess CTU n Prague, 2008 Fgure Subsystem wth S-Functon block...42 Fgure Mask of the subsystem...43 Fgure Smulnk model wth VR block...43 Fgure Mean solvng tme...47 Fgure Mean memory used...47 Fgure Rato of the solved nstances n tme...48 Fgure Smulnk model of DSVF...49 Fgure DSVF smulaton sgnals...50 Fgure Vrtual realty for the Host schedulng...51 Fgure Progress of the vsualzaton...51 Fgure Vsualzaton of the workshop...52 Fgure Motvaton example...53 Fgure A.1 - Generated Smulnk model...iv xv

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17 Dploma Thess CTU n Prague, 2008 Chapter 1 1. Introducton 1.1. The Vsualzaton n Schedulng Schedulng theory plays mportant role n optmzaton of resources that s used n many manufacturng and servce ndustres. Many optmal and heurstc algorthms have been proposed for the schedulng area, but there s a growng demand for transparent and realstc representaton of results n schedulng. The objectve of the vsualzaton s to brng these theoretcal results near to non-experts n schedulng theory. Especally producton schedulng and plannng needs to be represented n the transparent form. Vsualzaton s a technque for creatng mages, dagrams or anmatons for graphcal representaton of real systems. It has expandng applcatons n scence, educaton, engneerng (e.g. producton schedulng vsualzaton), nteractve multmeda, medcne, etc. Nowadays, the vsualzaton s used for graphc representaton of theoretcal background n the frst place. Ths representaton needs to be as precse real world approxmaton as t s possble. Therefore, vsualzaton should be desgned for concrete nstance of problem nstead of representng smlar problems by one predefned template. The best way to keep these demands s to bnd establshed schedulng notaton wth user-defned vrtual world defnton. Not only n relaton wth schedulng graphc representaton can be used. Another area wth wde use of the vsualzaton s presentaton of projects, ther results and motvaton examples. Transparent presentaton of a fnal soluton of any problem s very mportant. We can say almost as mportant as the soluton tself. Also motvatng students and ther approach to new areas of study can be supported by the vsualzaton. We can consder three basc areas of the vsualzaton use: 1) As a feedback for creatng a tme schedule. 2) For graphc representaton of already scheduled problems, e.g. for producton schedulng. 3) For demonstratonal purposes, e.g. a fnal soluton presentaton. 1

18 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek 1.2. Schedulng and the Smulaton The smulaton s used n many contexts, ncludng the modelng of natural systems or human systems n order to gan nsght nto ther functonng. Other contexts nclude smulaton of technology for performance optmzaton, safety engneerng, testng, tranng and educaton. Smulaton can be used to show the eventual real effects of alternatve condtons and courses of acton. Whereas vsualzaton has man purpose n the presentaton of schedulng results, smulaton can serve as an optmzaton tool. We can use smulaton not only n creatng tme schedules but also for optmzng system desgn, e.g. n dgtal processng. In both cases, smulaton can be used for cyclc optmzaton va modfcaton of some parameters by recalculaton wth acqured data or smply by estmaton. In comparson wth smulaton usng hardware, less tme s needed for a programme realzaton of the same problem. Desgn of dgtal flters can be mentoned as a sutable example. Easer modfcaton of whole system s also one of the advantages of the computer smulaton. In analog verson, some parameters of hardware elements can be changed to modfy functon of the system. Ths change may not guarantee needed precson of new value; on the other hand, change of parameters n dgtal verson s exactly defned and changes can be repeated nfntely. Consequently, we can say that computer smulaton s less vulnerable to errors caused by naccuracy of the project mplementaton. Consequently, we can descrbe basc areas of computer smulaton usage: 1) As an optmzaton tool for mprovng tme schedules. 2) For determnng the nfluence of gven schedule to whole system functon. 3) As an approxmaton of the real system and ts functon before hardware mplementaton. Ponts 2) and 3) are very closely related to desgn of dgtal sgnal processng unts, especally for dgtal flters. Usage of computer smulaton there leads to hgh reducton of tme needed to ther desgn. 2

19 Dploma Thess CTU n Prague, Optonal Schedulng Motvaton Schedulng theory tself assumes exactly gven set of tasks to be scheduled. Ths means that each gven task s present n the fnal schedule, only start tme and processor has to be assgned for each task. Nevertheless, there s at least one stuaton when classc concepton of tasks s not suffcent: problem wth alternatve process plans. In related works, there are also used terms alternatve (or optonal) tasks, actvtes and routngs [1], [2], [3], [4], [5]. In ths work, t wll be referred as the optonal schedulng problem or as the problem wth alternatve process plans. Not only for producton schedulng can ths stuaton occur, but producton schedulng s the most sgnfcant example from real world applcatons. Let us consder stuaton when there are more alternatve routes to satsfy all demands and constrants and each route s formed by dfferent set of tasks. Then not all of ntal tasks wll be present n the fnal schedule. We can say that presence of each task n the fnal soluton s only optonal, so the solvng algorthm has to be able to choose and schedule only subset of tasks nstead of creatng schedule from all of them. Presence of precedence constrants between tasks results from the prncple of the optonal schedulng problem. As mentoned above, ths stuaton occurs manly n producton schedulng because there are usually more ways how to complete the product. For example, to repar a televson one expert worker or two less sklled workers are needed. Frst alternatve for chef of the workshop s to assgn televson repar to one sklled worker who needs about 2 hours to complete the work and second alternatve s to commt t to two workers who need about 3 hours to do the same work. Each worker can use two dfferent analyzer unts where tme to fnsh the repar also dffers and these analyzer unts can represent shared resources for all workers. All these facts lead to more than one way how to complete the set of demands under fxed constrants. Much more examples of ths problem could be mentoned. In fact, the optonal schedulng problem s a combnaton of schedulng and plannng areas. Plannng conssts of sequencng of operatons to one ntegrated manufacturng plan. Only the operaton tself s consdered n the phase of creatng the producton plan ndependently on ts real processng tme and resource requrements. Precedence constrants are defned durng the plannng. Schedulng deals wth tasks specfcatons lke processng tmes, resource demands and precedence constrants. The goal of the schedulng phase s to create a tme schedule optmzng some crteron whle respectng all constrants. 3

20 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek 1.4. Contrbuton Ths dploma thess presents results n two areas of schedulng theory [6]. One part contans descrpton of the new tool for representaton of schedulng results va the smulaton and vsualzaton. Second part s dedcated to an optonal schedulng problem, ts descrpton, data representaton and an optmal algorthm for solvng of ths problem s proposed VISIS Applcaton Frst goal of ths dploma thess s to present an applcaton for demonstraton of schedulng results n the Matlab envronment ( : VISIS (VIsualzaton and SImulaton n Schedulng). Ths applcaton uses the Matlab-based smulaton envronment Smulnk and the Vrtual Realty toolbox for the graphc vsualzaton. Two areas of usage are consdered: smulaton for montorng of the nfluence of the schedulng process on the system functon (e.g. for dgtal flters) and tme vsualzaton (e.g. graphc representaton of the executon on a producton lne n tme). Ths tool wll be freely avalable n the next verson of TORSCHE Schedulng Toolbox for Matlab ( planed on October Up to our knowledge there s no such a tool provdng vsualzaton of scheduled processes n ths range. VISIS s able to smulate or vsualze any tme schedule gven by TORSCHE data structures. The applcaton s adjusted to maxmze user comfort and smplcty of usage. Therefore, most of data and fles needed for the smulaton and vsualzaton are created automatcally and the structure of the code that defnes actvtes s checked before the start of smulaton n Smulnk. Vrtual realty world s fully user-defned so t can satsfy all appearance demands. In addton, parameters that can be changed va control code are defned by user of the applcaton. Vsualzaton can be realzed n 2D or 3D world. Some examples were created as an llustraton of the VISIS potental. All data hstory can be saved to later analyss and more, Vrtual Realty toolbox used for vsualzaton provdes possblty to capture any frame or record runtme as a vdeo fle. Smulnk allows stoppng and restartng the smulaton n any tme and t s also possble to change some parameters durng the smulaton. VISIS s desgned to provde transparent representaton of schedulng results n the frst place but t can be also used for optmzaton purposes. Especally smulaton usng VISIS can 4

21 Dploma Thess CTU n Prague, 2008 serve as a fast feedback to acqured schedule and t can contemporaneously show functon of whole smulated system. Ths can be great help for solvng problems wth assgnng computng operatons for multprocessor systems and also for desgnng dgtal processng unts at all Related Works for the Vsualzaton and Smulaton Ths work comes up from TORSCHE Schedulng Toolbox for Matlab [7], whch provdes data structures and algorthms for tme schedulng. Thus, all work s realzed n the Matlab envronment. One of the related tools s TrueTme [8] - a Matlab based tool for realtme smulaton for wde spectrum of problems, e.g. dgtal flters, embedded systems or wreless networks. TrueTme s very strong smulaton tool but utlzaton for small problems s qute dffcult and especally for schedulng results smulaton t s unnecessarly sophstcated soluton. Fshman [9] made a comprehensve revew of Dscrete Event Smulaton (DES) whch s a system represented by sequence of dsjunctve events. Each event occurs at an nstant tme and leads to change n the system state whle there are not contnuous states. Therefore, DES systems are very closely related to the smulaton of schedulng results because each task s one dscrete event, although t can contan more operatons. But only the begnnng and the end of each task s mportant for the smulaton of a tme schedule. Optmzaton usng smulaton s also descrbed n ths book. Another utlzaton of smulaton-based optmzaton n real producton was shortly descrbed by Manlg and Sramek [10]. An alternatve for purely sequental dscrete event smulaton was proposed by Msra [11]. Ths survey s dedcated to dstrbutng of DES to more cooperatng processors what may provde better performance of smulaton. Utlzaton of smulaton for producton schedulng was dscussed by Toal, Coffey and Smth [12], ncludng also expert systems utlzaton. For vsualzaton purposes, OpenGL (Open Graphcs Lbrary) ( s a standard specfcaton defnng a cross-platform API for wrtng applcatons that produce 2D and 3D computer graphcs. Ths means that vsualzaton can be realzed by OpenGL at any operaton system. The vsualzaton n schedulng was studed at Karlsruhe Unversty [13] and some applcaton for vsualzaton of process schedulng has been developed there. Ths tool has predefned template where up to 8 processes (tasks) wth some attrbutes and up to 4 processors can be defned. Tme schedule s then represented by showng state of all tasks and processors n tme. On the other hand, Matlab ncludes Vrtual Realty toolbox, whch s 5

22 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek also suffcent tool for vsualzaton of schedulng results. More tools can be mentoned but none of them provdes vsualzaton n close relatons wth user-defned tme schedules Algorthm for the Optonal Schedulng Problem Second part of ths thess s dedcated to descrpton of the new schedulng algorthm for the problem wth alternatve process plans. Ths nvolves not only realzaton of algorthm tself but also nput data representaton and a specfcaton of the problem types that can be solved. Schedulng algorthm s based on Integer Lnear Programmng (ILP) [14] and s mplemented n the Matlab envronment usng addtonal ILP solver. The formalsm of Petr nets [15] n combnaton wth TORSCHE toolbox structures and algorthms s used for nput problem settng and solvng. Soluton for a certan area of schedulng problems, ncludng case wth alternatve process plans and mnmzng C max crteron n the frst place, was mplemented. Moreover, ILP algorthm s able to solve problems wth gven processng tmes, release tmes, deadlnes and precedence constrants for tasks whle one processor, dedcated processors or nfnte amount of dentcal processors are avalable. We can say that each problem denoted by ~ { 1, P, PD} p, r, d C [6] that can be nterpreted va Petr nets s solvable. The proposed max soluton can be easly modfed to optmzng another crteron that can be descrbed as a lnear combnaton of tasks start tmes and processng tmes. Defnton of the problem structure usng PN notaton allows easy modfcaton of the created project va smple graphc nterface. Output of the applcaton s the standard TORSCHE toolbox structure taskset wth the ncluded tme schedule that can be depcted as a Gantt chart [6]. It s also possble to use VISIS applcaton to vsualze or smulate acqured soluton Related Works for the Optonal Schedulng Problem There were some attempts to deal wth the optonal schedulng problem but no one n the same form as descrbed n ths work. Bartak [1] proposed edge-fndng algorthm for schedulng wth optonal actvtes (tasks) wth unary resource. Ths algorthm s able to solve 6

23 Dploma Thess CTU n Prague, 2008 the problem wth gven processng tme, earlest possble start tme (release tme) and latest possble completon tme (deadlne) for each task. The functon of the algorthm s based on tghtenng the tme bounds for tasks and ther progressve elmnaton from the ntal set of tasks. Beck and Fox [2] formulated a constrant-based representaton of optonal actvtes to model problems contanng alternatves n schedulng. They begn wth lstng of all possble schedulng alternatves and then they connect these dfferent ways nto one temporal graph wth so-called XOR nodes. Each of the XOR nodes represents one branchng to more alternatves where decson has to be made. Beck and Fox assgn each task wth probablty of exstence (PEX) value, bounded wthn nterval 0, 1. Rules for propagaton of these values through the graph are then defned and rules for connecton of graph nodes are also descrbed. Exact workng algorthms and some heurstcs are proposed for soluton of problems set by ths graph. Ths representaton by the graph wth establshed XOR nodes s n very close relaton wth Petr nets formalsm used n ths dploma thess. Wlhelm [16] proposed column model represented by state transton graph to deal wth assembly system desgn problem wth tool changes. Each node n the graph represents one staton wth sngle operaton assgned and determned tme needed for ths operaton. Algorthm for soluton of ths problem s then based on fndng the shortest path through the graph respectng gven constrants. Helmann [17] establshed model for project schedulng wth mult-mode resources where each actvty can be realzed on more than one resource whle processng tmes and also mnmum and maxmum tme lags for other tasks dffer. Soluton s set on branch and bound method wth depth-frst search. Utlzaton of Petr nets n schedulng area was descrbed by Tuncel and Bayhan n [18]. They dscuss applcaton of Petr nets n producton schedulng and then they descrbe combnaton of PN formalsm wth search algorthms, heurstcs, meta-heurstcs and mathematcal based algorthms lke lnear programmng for mnmzng the cycle tme of the system. Algorthm for mnmzng total tardness n a flexble manufacturng system based on Petr nets formalsm was proposed by Meja and Montoya [19]. They use transtons to represent events (tasks) and places to represent states, condtons and machnes (processors). Ths approach s very close to the problem representaton descrbed n ths thess except the processors representaton. Soluton of the problem s based on state equatons of the system and a heurstc search algorthm s also descrbed. 7

24 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek 1.5. Paper Organzaton The paper s dvded nto eght chapters. Frst chapter presents motvaton of ths thess and ts contrbuton n two areas: smulaton and vsualzaton of schedulng results n the frst place and optonal schedulng as second. Related works are also descrbed there. Chapter 2 provdes basc schedulng theory descrpton, then some examples of schedulng problems are descrbed and representaton of ther results s dscussed. Statement of the optonal schedulng problem s provded at the end of ths secton. Next chapter s dedcated to descrpton of the optonal schedulng problem representaton and ts relaton to Petr nets formalsm. Chapter 4 descrbes Integer Lnear Programmng formulaton for the optonal schedulng problem. Chapter 5 descrbes areas of utlzaton of both vsualzaton and smulaton tools. Followng chapter s dedcated to descrpton of mplementaton of VISIS applcaton n Matlab, ts relaton to TORSCHE toolbox and short descrpton of created functons and other fles. Chapter 7 presents experments, case studes and examples for both parts of ths dploma thess. Last chapter contans conclusons of ths work, contrbuton to schedulng area and future utlzaton of VISIS applcaton and the optonal schedulng algorthm. 8

25 Dploma Thess CTU n Prague, 2008 Chapter 2 2. Problem Descrpton 2.1. Schedulng Theory Schedulng s an optmzaton of resources usage wth gven constrants n tme. In other words, schedulng solves the problem how to assgn gven resources to gven tasks n tme. Schedulng problems are characterzed by three basc sets [6]: 1) Set of tasks - T = { T,, } 1 K 2) Set of processors (machnes) - P = { P,, } T n 1 K 3) Set of addtonal resources - R = { R,, } 1 K Tasks are determned by several numercal parameters, e.g. processng tme. There are two general constrants: each task can be processed by at most one processor at a tme and each processor s able to process at most one task at a tme. Set of processors P determnes amount and type of processors that can be used. Resources needed for executon of tasks, whch are not ncluded n set P, are represented by the set of addtonal resources R. For determnstc problems, α β γ notaton was establshed [6], [20]. The frst feld α descrbes avalable resources (processors), β represents tasks and resources characterstcs and γ denotes an optmalty crteron. Detaled descrpton s below. Many algorthms were proposed for soluton of problems descrbed by ths notaton. R s P m Generally, schedulng s NP-hard problem. Polynomal algorthms are known only for the lmted amount of schedulng problems, especally for problems wth only one processor. For the rest of the problems, solvng tme needed to fnd optmal soluton s exponentally proportonal to sze of the nput problem. Algorthms can be based on searchng technques lke branch and bound method, on constrant programmng [21] or on nteger lnear programmng [14]. Other way s to use some polynomal heurstc method but wthout assurance of optmal soluton. Producton schedulng s a branch of schedulng mostly amed to automated producton lnes and ndustral producton at all [22]. Almost all cases of producton schedulng are NPhard problems. 9

26 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Defnton of the Fundamental Schedulng Objects There are some essental notons from schedulng theory used n ths work: Task - Basc schedulng object, referred to as T. - Represents one concrete real world operaton. - Each task has ts own executon tme called processng tme p. - May have some addtonal propertes: Release tme r, whch s the tme at whch T s ready for processng. Due date d, whch s the tme lmt by whch T should be completed. Deadlne d ~, whch s a tme lmt by whch T must be completed. Weght w, whch represents relatve urgency of task T. Resource specfcaton: on whch resource has to be T processed. - Scheduled task has ts own start tme s and completon tme c n the schedule. All parameters p, r, d, d ~ and w are supposed to be postve ntegers. Taskset - Set of tasks T, referred to as T. - All tasks have the same resource envronment. Job - Set of tasks T, referred to as J. - Each task has ts own resource specfcaton - dedcated processors problem. Problem - Descrpton of tasks, resources and constrants. - Establshed α β γ notaton was descrbed by Blazewcz [6]. α characterzes the type and amount of processors used: Ø - one processor. P - dentcal processors. Q - unform processors. R - unrelated processors. PD - dedcated processors, defnton proposed by Kellerer and Strusevch [23]. O - dedcated processors, open shop system. F - dedcated processors, flow shop system. 10

27 Dploma Thess CTU n Prague, 2008 J k - dedcated processors, jobshop system. - number of processors; nteger value, wrtten after type. β descrbes tasks and resources characterstcs: pmtn prec p, r, d, d ~ - preempton allowed. - precedence constrants between tasks exsts. - detaled descrpton of processng tmes, release tmes, due dates and deadlnes. γ denotes the optmalty crteron; some of the most used crterons: C max - mnmzng of the latest completon tme. Represents throughput of the system. C w - mnmzng of the weghted sum of completon tmes. U - mnmzng of the number of delayed tasks. L max - mnmzng of the largest L value, where L = c d. Schedule - Result of the schedulng. - Descrbes allocaton of tasks to resources n tme. - Mostly represented by Gantt chart [6] wth tme on x-axs and resource on y-axs Examples of Schedulng Problems In ths subsecton, some examples of the schedulng problems are shown. ~ 1 r, d C max - problem wth one processor, gven processng tmes, release tmes and deadlnes for tasks. Optmalty crteron s to mnmze the latest completon tme. Let us have a set of four tasks, determned by the vector of processng tmes p = [2,1,2,2 ], the vector of ~ release tmes r = [4,1,1,0 ] and the vector of deadlnes d = [7,5,6,4 ]. Graphcal representaton of ths problem va modfed Gantt chart s shown n Fgure 2.1. Ths pcture was acqured usng TORSCHE toolbox. Each task has ts own processng tme (wdth of approprate rectangle area), release tme (arrow amng up) and deadlne (arrow amng down). 11

28 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Input set of tasks Task 1 Task 2 Task 3 Task t Fgure r, ~ d Cmax problem P2 prec - problem wth two dentcal parallel processors, gven processng tmes, due L max dates and defned precedence constrants between tasks. Presence of due date values s not nvolved n problem defnton, but t s a logcal consequence of the crteron L max, whch s subject to due dates. Optmalty crteron s to mnmze the largest L, where L = c d. Value L max can be also negatve - n the stuaton when all tasks are completed before ther due dates. Precedence constrants can be defned by the task-on-node graph [6], shown n Fgure 2.2. Each node of ths graph represents one task and each edge denotes a precedence constrant between approprate tasks. Fgure Precedence constrants 12

29 Dploma Thess CTU n Prague, Gantt Chart As mentoned above, the most used representaton of tme schedules s Gantt chart. All task parameters and precedence constrants can be dsplayed n one chart. There are dscrete tme values on x-axs and descrpton of processors on y-axs. Each start tme of the task s represented by shft of approprate task poston along x-axs and assgnment of the task to ~ processor s determned by poston on y-axs. Let us take an example of 1 r, d C problem from the prevous subsecton. Gantt chart resultng from the schedulng process, for example usng branch and bound method, s shown n Fgure 2.3. max Scheduled tasks Processor1 T 4 T 2 T 3 T t Fgure Scheduled problem 1 r, ~ d Cmax Problem max P 2 prec L from the prevous subsecton s taken as an example of multprocessor Gantt chart. Taskset characterstcs: set of processng tmes p = [2,2,2,3,3,3 ] and set of due dates d = [3,5,8,6,4,3 ]. Resultng schedule wth precedence constrants taken from Fgure 2.2 s dsplayed n Fgure 2.4. Scheduled tasks Processor1 T 1 T 5 T 4 T 3 Processor2 T 6 T t Fgure Scheduled problem P2 prec Lmax 13

30 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek The Host Schedulng Problem The host schedulng problem [24], [25] deals wth the problem how to schedule the host moves to perform the materal handlng tasks n the system. The most frequent used optmalty crteron s C max value because t represents throughput of the system. In the host schedulng problem, each task represents one move of the host wth the materal. The materal has to be processed n several tanks wth lqud and the tme needed for processng n every tank s determned by ts mnmum and maxmum value. In addton, all empty host moves have to be taken nto account for the schedulng. Result of schedulng depcted by Gantt chart s shown n Fgure 2.5 and representaton by a specal chart for the host schedulng problem s shown n Fgure 2.6. Host 1 T 1 T 3 T 2 T 4 T t Fgure Host schedulng Gantt chart T = tanks t Fgure Host schedulng specal chart Specal chart for the host schedulng problem gves better dea of the acqured result, although understandng s qute dffcult for the frst tme. Red sold lnes n Fgure 2.6 represent moves of the host wth the materal, red dashed lnes depct empty host moves and blue lnes represent temporary stays of the materal n tanks. Labels of tanks are on y-axs and dscrete tme s on x-axs. 14

31 Dploma Thess CTU n Prague, Optonal Schedulng Problem Statement In contrast to classc concepton of tasks descrbed n the prevous secton, optonal schedulng problem s a problem that nvolves dfferent approach for solvng. More possble process plans are defned and only one of them has to be chosen. Process plan s a sequence of tasks that satsfes all nput demands. All tasks can be ncluded n more than one plan, but each sequence s determned by a dsjunctve set of tasks,.e. no task can be ncluded n one process plan more than once. We can say that each process plan represents one alternatve routng to complete the nput assgnment. Not all gven tasks wll be then present n the fnal schedule, so they can be called optonal tasks [1]. These tasks have the same propertes as descrbed n Secton 2.1 and also processors have the same characterstcs. The only dfference s that only one process plan has to be chosen and scheduled, so only one partcular sequence of tasks s taken nto account for schedulng. Ths s very close to combnaton of plannng and schedulng areas, but all decsons are made only on the bass of the schedulng algorthm. No addtonal nformaton except the defnton of tasks and alternatve process plans has to be set before the schedulng process. The goal of the schedulng for the optonal schedulng problem s to choose one process plan and assgn all ncluded tasks to processors accordng to the gven crteron and respectng all gven constrants. Result s then classc tme schedule contanng selected tasks. Algorthm for the optonal schedulng problem has to be dfferent from classc schedulng algorthms from the pont of vew that not all constrants resultng from the problem assgnment have to be satsfed. For example, f one task s ncluded n two dfferent process plans, then only precedence constrants from the chosen plan are taken nto account for the schedulng process. Another problem s wth task start tme n the schedule: f all plans would be scheduled, the same task n dfferent plans could have dfferent start tme. Ths s n contradcton wth the fact, that each task can have only one start tme for determnstc problems. 15

32 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Chapter 3 3. Optonal Schedulng Problem Representaton In order to develop a new algorthm for the optonal schedulng problem, the nput data representaton has to be defned frst of all. Two general demands have to be satsfed. Proposed data representaton must be able to descrbe more alternatve process plans n one nstance of a problem and data entry should be comprehensble and should allow easy modfcaton. For ths purpose, exstng related work s used as the startng pont and the proposed soluton s then modfed usng the Petr nets formalsm Modfed Temporal Network - XOR Graph As mentoned n the prevous secton, the optonal schedulng problem nvolves approach dfferent from the stuaton when all gven tasks are present n the fnal schedule. For ths purpose, we wll come out of the model proposed by Beck and Fox [2] and then ths model wll be modfed and nterconnected wth the Petr nets formalsm. Beck and Fox start wth lstng of all alternatve process plans as a dsjunctve set of task consequences, shown n Fgure 3.1 (taken over from [2]). Fgure Alternatve process plans Each task has ts own dentfer, wrtten n the lower-rght corner of the box, and resource to be processed on, wrtten n the upper-left corner of the box. Each process plan (PP) s determned by a consequence of dsjunctve tasks. The goal of the schedulng process s then to choose one of those process plans and create a tme schedule of ncluded tasks respectng ther characterstcs, resources specfcaton and precedence constrants. As we can see n Fgure 3.1, each task can be ncluded n more than one process plan, but each process 16

33 Dploma Thess CTU n Prague, 2008 plan s determned by a dsjunctve set of tasks. Task presence n the fnal soluton s determned by a probablty of exstence - PEX value. As a next step, all process plans are connected nto one graph - modfed temporal network, shown n Fgure 3.2. Ths graph, also called XOR graph, conssts of three types of nodes: 1) Actvty node s representaton of task. Start tme and completon tme of an Actvty node n the fnal schedule are temporal varables. Duraton of an actvty node n the temporal graph s determned by the processng tme of an approprate task. 2) AND node has start tme and completon tme n the schedule and these values are temporal varables. Duraton of AND node s zero. For the PEX value propagaton, two rules are defned: 1) all graph nodes lnked to an AND node exst n the soluton f and only f the AND node does; 2) All non-xor nodes drectly connected to an AND node must have the same PEX value as the AND node tself. 3) XOR node represents the possblty of choce durng the schedulng process. Duraton of a XOR node s zero and ts start tme and completon tme are temporal varables agan. Rules for PEX propagaton through the XOR nodes are these: 1) At most one node connected to a XOR node upstream and one connected downstream can be present n the fnal schedule; 2) If there s a node connected to the XOR node downstream (upstream), whch s present n the fnal schedule, then there must be just one node connected to the XOR node upstream (downstream) that s present n the fnal schedule and also XOR node tself must be present n the schedule. Fgure 3.2 (taken over from [2]) dsplays XOR graph resultng from the problem assgnment shown n Fgure 3.1 where four dfferent process plans are defned. All these plans are nterconnected by one XOR node at the begnnng and one at the end. Moreover, plans 3 and 4 have the same frst actvty (task), so we can use branchng va XOR node up to ths actvty. Smlarly, the last actvtes of plans 3 and 4 are also the same, so we can connect these plans va XOR node before ths actvty node. 17

34 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Fgure XOR graph 3.2. Petr Nets n Optonal Schedulng In the followng text, we modfy XOR graph presented n the prevous secton and then we use the Petr nets formalsm [15] to set the nput problem. For ths purpose, the Petr nets formalsm has to be defned n the frst place Petr Nets Bascs A Petr net s a formalsm for modelng of systems wth dscrete states and events. Petr net (PN) s a quntuple,, F, W, M } { T,, } 1 K T n { 0 P T, where P { P,, } = s the fnte set of places, T = s the fnte set of transtons, F ( P T ) U ( T P ) s the set of arcs, 1 K P m W : F {1, 2, K} s a weght functon, M : P {0, 1, 2, } s the ntal markng [15]. P and 0 K T are dsjunctve,.e. no object can be both a place and a transton. Petr nets are represented by a bpartte drected graph. Every arc of the graph connects one place and one transton; arc cannot connect two nodes of the same type. Consequently, places and transtons are regularly alternatng n a graph. Transton wthout nput place s called source transton and transton wthout output places s called snk transton. The same stuaton occurs for places. For the purpose of modelng and smulatng of systems, each place can contan any number of tokens. A dstrbuton of tokens over the places of a Petr net s called markng M. An example of a smple Petr net contanng three places and three transtons s depcted n Fgure 3.3. Markng of ths net s = ( 1, 0, 0) M. 18

35 Dploma Thess CTU n Prague, 2008 Fgure Example of Petr net The most mportant parameters of a Petr net for schedulng are state-transtons matrces W and W and resultng ncdence matrx W. W s defned as of places and n s amount of transtons. Columns of matrx m n matrx, where m s amount W represent transtons and rows represent places. Each element of a matrx determnes number of tokens taken from a place by an approprate transton. Smlarly, + W s defned as m n matrx where each element of a matrx determnes number of tokens added to a place by an approprate transton. Incdence matrx of a Petr net s then defned as W = W W +. Generally, arcs between places and transtons can have dfferent weghts but for the optonal schedulng problem purposes, all arcs are supposed to have weght equal to one. Also markng s gnored n that case. + State-transton matrces resultng from the Petr net depcted n Fgure 3.3: W 1 = W + 0 = Incdence matrx: W = W + W 1 = There are some specal PN structures that are mportant for the optonal schedulng problem. Possblty of choce, needed for the data settng, s represented by a place wth more 19

36 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek output transtons. Smlarly, mergng of more alternatve process plans s modeled as a place wth more nput transtons. Only one of tasks T 1, T 2 and T 3 can be present n the soluton of problem depcted n Fgure 3.4. Fgure Choce representaton n PN Stuaton wth a process plan contanng tasks that can be executed smultaneously s represented by a transton wth more output (and then nput) places (see Fgure 3.5). Both tasks T 2 and T 3 have to be scheduled and they can be executed n the same tme f there s suffcent resource envronment. Fgure Parallel process plans n PN Not all models that can be defned va Petr nets are correct representatons of the real process plans. There are some cases nconsstent wth the dea of the process plans; one ncorrect nstance of Petr net s shown n Fgure 3.6. The problem s n the fact that there s a branchng nto two alternatves and then these dsjunctve parts are connected by one transton. Ths s classc example of deadlock because n the begnnng, only one of two alternatves has to be chosen and later, both process plans need to be executed to reach the fnal state. 20

37 Dploma Thess CTU n Prague, 2008 Fgure Incorrect PN Converson of XOR Graph to Petr Net In our case, transtons stand for actvtes (tasks) and places stand for the states of the system. In comparson wth XOR graph, transtons represent both AND nodes and Actvty nodes and places represent XOR nodes. Ths approach respects all rules for nodes n XOR graph defned n the prevous text. All places connected wth a transton are present n the fnal soluton f and only f the transton s also present. For each place that s present n the fnal soluton, there s just one nput transton and one output transton, except the place representng ntal state of the system (no nput transtons) and place representng the fnal state of the system (no output transtons). Wth ths presumpton, we can modfy the optonal schedulng problem depcted n Fgure 3.1 and Fgure 3.2 to the verson based on the Petr nets formalsm (see Fgure 3.7). 21

38 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Fgure PN for the optonal schedulng problem As mentoned above, possblty of choce durng schedulng s modeled by a place wth more output (and later nput) transtons. At the begnnng, only one of the tasks A 1, A 3 and C 1 has to be chosen and approprate process plan has to be scheduled then. In the stuaton when C 1 s chosen, another one decson has to be made. Input representaton of the optonal schedulng problem s defned as follows: 1) Each task s represented by a transton n a gven Petr net. 2) Alternatve process plans are modeled by a place wth more output (and later nput) transtons. 3) Parallel tasks are modeled by a transton wth more output (and later nput) places. 4) Petr net for the optonal schedulng problem must have one source place, representng the start pont for schedulng, and one snk place, representng the requested fnal state. 5) No source or snk transtons are allowed. 22

39 Dploma Thess CTU n Prague, 2008 Any PN edtor that provdes possblty to obtan the state-transtons matrces + W and W can be used for the nput problem settng. Descrpton of the Petr net by the statetranston matrces s passed to the proposed schedulng algorthm. Tasks are determned by ther processng tmes, release tmes, deadlnes and resource specfcaton f needed. All these parameters are passed to the algorthm as well. 23

40 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Chapter 4 4. Algorthm for the Optonal Schedulng Problem In ths secton, an optmal algorthm based on Integer Lnear Programmng (ILP) for the optonal schedulng problem s proposed. We use the Petr nets formalsm descrbed n the prevous secton to defne the structure of the alternatve process plans. State-transtons matrces + W and W are passed as nput data to the algorthm. GLPK (GNU Lnear Programmng Kt) solver [26] s used to solve the ILP problem. GLPK package s ntended for solvng large-scale lnear programmng (LP) and mxed nteger programmng (MIP). It s a set of routnes wrtten n ANSI C and organzed n the form of a callable lbrary. Ths tool s freely avalable at Defnton of Parameters and Varables The goal of the algorthm s to chose and then schedule a subset of the ntal set of tasks T = { T,, } 1 K T n, represented by dentcal set of transtons. Each task s determned by ts processng tme p and the structure of the problem s defned va Petr net. Deadlne d ~, release tme r and dedcated processor number R can be also defned. Poston of a task n the schedule s determned by ts start tme s. Presence of a task T n the fnal schedule s determned by a bnary decson varable v where v = 1 f T s present n the fnal soluton and v = 0 otherwse. For the purpose of ILP formulaton for the optonal schedulng problem, a bnary decson varable e,j s defned as presence of arc between place T and transton P j n the fnal soluton, e 1 f an approprate arc s present n the fnal soluton, j = and e 0 otherwse. Further, let x,j be a bnary decson varable such that x 1 f and, j =, j = only f s + p s j (.e. T s followed by T j ) and x, j = 0 f and only f s s j + p j (.e. T j s followed by T ) [27]. Objectve of the algorthm s to mnmze C max defned as C = max( s + p ). max 24

41 Dploma Thess CTU n Prague, 2008 From a defned Petr net, only state-transtons matrces + W and W are taken nto account for schedulng. Each element of the resultng matrx W determnes number of tokens removed or added to a place (ndex of a row) by an approprate transton (ndex of a column). In our case, all elements are from set { 1,0,1} because all arcs n a gven Petr net must have value equal to one. W [, j] = 1 denotes that one token s added to place P by transton T j, W [, j] = 1 denotes that one token s removed from place P by transton T j and W [, j] = 0 means that there s no arc between place P and transton T j. Another parameter, placetype, s computed from the matrx W. Ths parameter s a set wth the same sze as the set of PN places P and determnes the type of each place; placetype = 1 denotes that place P s a snk place, placetype = -1 sgnfes that place P s a source place and placetype = 0 denotes that place P has both nput and output transtons. Lst of all parameters and varables used n the ILP formulaton s summarzed below. Parameters: n - number of tasks; also number of transtons n a Petr net. m - number of places n a Petr net. UB - suffcently hgh postve nteger constant. { T,, } T = - set of tasks; equal to set of PN transtons. 1 K T n { P, } P =, - set of PN places. 1 K P m + W = W W - ncdence matrx of a Petr net; sze of W s m n. R = R, K, R ]- numbers of processors for dedcated processors problem. [ 1 n p = p, K, p ]- vector of processng tmes. [ 1 n r = r, K, r ] - vector of release tmes. [ 1 n ~ ~ ~ d = [ d, K, d ] - vector of deadlnes. 1 n placetype = placetype, K, placetype ] - ndcaton of the source and snk places of a [ 1 m Petr net. 25

42 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Varables: s - start tme of task T, = 1, 2, K, n ; nteger varable. v - presence of task T n the fnal schedule; = 1, 2, K, n ; bnary varable. e,j - presence of arc between place P and transton T j n the fnal schedule; = 1, 2, K, m ; j = 1, 2, K, n ; bnary varable. x,j - allocaton of tasks T and T j n the tme; = 1, 2, K, n, j = 1, 2, K, n ; bnary varable. - x 1 means that T precedes T j., j = - x 0 means that T j precedes T., j = Cmax - Value of the crteron; latest completon tme Integer Lnear Programmng Formulaton Let us descrbe basc dea of the proposed ILP formulaton for the optonal schedulng problem. Three smlar ILP models are descrbed, one for the problem wth one processor, one for nfnte amount of parallel processors and the last one for the problem wth dedcated processors. The goal of the ILP formulaton s to chose and then schedule a subset of the set of tasks T subject to C max crteron One Processor ILP model for the problem wth one processor s descrbed n the frst place: s + ( 1 v ) UB r = 1, 2, K, n (1) s ~ d p + (1 v ) UB = 1, 2, K, n (2) p j s k s j + UB ( 2 v j vk ) = 1, 2, K, m; j, k = 1, 2, K, n; W[,j] > 0 and W[,k] < 0 (3) p j s s j + UB x + UB (2 v v, j j ), j = 1, 2, K, n; k = 1, 2, K, m; < j and W[k,] > 0 and W[k,] ± W[k,j] (4) 26

43 Dploma Thess CTU n Prague, 2008 s s j + UB x UB p + UB (2 v v, j j ), j = 1, 2, K, n; k = 1, 2, K, m; < j and W[k,] > 0 and W[k,] ± W[k,j] (5) ( e, j ) = ( e, k ) j= 1,2K, n : W [, j] < 0 k = 1,2K, n : W [, k ] > 0 placetype = 1, 2, K, m (6), = = 1, 2, K, n; j, k = 1, 2, K, m; W[k,] > 0 and W[j,] < 0 (7) e k e j, = = 1, 2, K, n; j = 1, 2, K, m; W[j,] 0 (8) v e j, s + p C + UB 1 v ) = 1, 2, K, n (9) max ( We start wth two equatons (1) and (2) representng the constrants resultng from release tmes and deadlnes of tasks. If task T s not present n the fnal schedule (v = 0) then the constrants (1) and (2) are satsfed due to hgh postve constant UB and value of s s arbtrary. If v = 1 then the task cannot start before ts release tme and must be completed before ts deadlne. Precedence constrants between tasks are taken nto account n equaton (3). Ths equaton s created for all pars of transtons (tasks) T j and T k such that the output place P of the transton T j s the nput place of the transton T k. Constrant for start tmes of tasks s consdered only f both tasks are present n the fnal schedule,.e. v = 1 and v = 1, constant UB ensures satsfacton of the equaton otherwse. Equatons (4) and (5) k represent processors constrants for all pars of tasks that are not joned by precedence constrants. These equatons ensure that only one task s processed by the processor n a tme. Constrant (6) represents lmtaton for number of nput and output arcs that can be n the fnal schedule for each place. Number of output arcs present n the fnal soluton s equal to number of nput arcs present n the fnal schedule. Parameter placetype ensures that just one output arc of the source place and one nput arc of the snk place wll be present n the fnal soluton and therefore at most one nput/output arc can be present n the fnal soluton for each place n a Petr net. Constrant (7) denotes that all nput and output arcs of each transton have the same value of varable determnng ther presence n the fnal soluton. Equaton (8) determnes that the transton T s present n the fnal soluton f and only f all nput and j 27

44 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek output arcs are also present. Equaton (9) stands for evaluaton of optmalty crteron C max ; only completon tmes of tasks that are present n the fnal schedule are taken nto account Infnte Amount of Identcal Parallel Processors ILP model, consderng nfnte amount of dentcal parallel processors, s dentcal to the ILP model wth one processor. Only processors constrants are not taken nto account, so the equatons (4) and (5) are not consdered Dedcated Processors The problem wth dedcated processors s close to the problem wth one processor, mentoned n the prevous text, except the processor constrants. Therefore, equaton (4) and (5) are modfed to equatons (10) and (11). Equatons themselves are the same as n the case wth one processor, but they are created only for tasks assgned to the same processor R. p j s s j + UB x + UB (2 v v, j j ), j = 1, 2, K, n; k = 1, 2, K, m; < j and W[k,] > 0 and W[k,] ± W[k,j] and R = R j (10) s s j + UB x UB p + UB (2 v v, j j ), j = 1, 2, K, n; k = 1, 2, K, m; < j and W[k,] > 0 and W[k,] ± W[k,j] and R = R j (11) 28

45 Dploma Thess CTU n Prague, 2008 Chapter 5 5. Utlzaton of the Smulaton and Vsualzaton Areas of the smulaton and vsualzaton use n schedulng wll be descrbed n ths secton and some examples wll be shown. Basc summary of ths theme was shortly mentoned n Chapter 1. Both smulaton and vsualzaton have to be desgned just for each case separately. There are some examples to llustrate utlzaton of the smulaton and vsualzaton n the followng text. Some concrete mplemented problems wll be shown n Chapter Applcaton of the Smulaton Schedulng of dgtal sgnal processng (DSP) algorthms s a practcal example where the smulaton can be used. Let us slghtly descrbe the functon of the Dgtal State Varable Flter ( [28], [29] that s used for example n processng of acoustc sgnals. Ths flter arses from the authentc analog verson and ts transcrpton results n the set of mathematcal equatons that represent flterng operatons. Functon of the flter s then realzed by repeatng of those operatons n a neverendng loop. One of the reasons for the utlzaton of the dgtal verson s easer desgn of the flter and easer modfablty as well. Role of the schedulng n the desgn of dgtal processng unts, ncludng flters, s to allocate gven operatons to one or more processors n tme. From the schedulng process pont of vew, each mathematcal operaton correspondng to one task T and tme needed for executon of ths operaton corresponds to processng tme p. The goal of the schedulng algorthm s to create a tme schedule contanng all tasks whle optmzng the gven crteron. In the case of dgtal flter desgn, objectve s to mnmze the cycle tme (perod) of the schedule [27], [29]. Dgtal state varable flter s formed by the set of equatons that are summarzed below. for k = 2 : K FB ( k) = F1 B( k 1) // T 1 L ( k) = L( k 1) + FB( k) // T 2 QB ( k) = Q1 B( k 1) // T 3 IL{ k} = I( k) L( k) // T 4 29

46 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek H ( k) = IL( k) L( k) // T 5 FH ( k) = F1 H ( k) // T 6 BK = FH ( k) + B( k 1) // T 7 N ( k) = H ( k) + L( k) // T 8 end Appled attrbutes: k - current teraton of the algorthm. K - number of teraton. I - nput value. L - output value. F 1, Q 1 - constants. FB, QB, IL, H, FH, B, N - dscrete states; ntally equal to zero. Each mathematcal operaton s assgned to one task whereas addton takes one clock cycle of the processor unt and multplcaton takes three clocks. Precedence constrants result from the relatonshp of operatons used n the equatons (see Fgure 5.1). Fgure Precedence constrants for DSVF flter Two dscrete tme moments are mportant for the schedulng process for each task. Frst of them s the begnnng of the task when all nput data for an approprate equaton are fetched and the second one s the end of the task when all computed data are uploaded to the output (see Fgure 5.2). Some computaton s executed between these moments and data are beng updated. 30

47 Dploma Thess CTU n Prague, 2008 Fgure Tasks executon Each operaton (task) must be executed just once durng one teraton of the algorthm. Result of the smulaton of DSVF flter wth sample tme equal to 220 khz s dsplayed n Fgure 5.3. The result was acqured usng TrueTme [8] tool n Matlab. Fgure Result of the smulaton Not only flterng of sgnals can be mentoned as utlzaton of the smulaton n dgtal sgnal processng. Model-based Predctve Control (MPC) [30], [31] s another representatve example of optmzatons n DSP. MPC controllers are ntended for control of lnear (or lnearzed) systems subject to optmzaton of specfed optmalty crteron. An advantage of MPC control s an occason to cover up wde spectrum of constrants and restrctons lke lmts for the value of control sgnals or for ther rate of change n tme. The most frequently used demand s to observe the reference sgnal. Desgn of MPC controllers arses from the state space equatons of the system. The optmalty crteron s formed by lnear combnaton of energy used to control and squared dfference between the requred and real output. The goal of the MPC controller s to fnd a sequence of dscrete values of control sgnal mnmzng the crteron. Number of computed control sgnal values s called predcton horzon. There are more control strateges wth MPC controller. One strategy s to compute the control sequence only on bass of state space model of the system and then to apply ths control wthout regard to actual state of the system. Other strategy, closer to applcaton of the smulaton, s based on recalculaton of the control sequence after each dscrete step n tme. 31

48 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Ths strategy s called recedng horzon because n each recalculaton, new predcton horzon s computed, so the end of predcton s beng shfted. Soluton of the MPC control problem can be found n numercal verson, formed by the set of mathematcal operatons that represent matrces operatons. These equatons can be separated to elementary mathematcal operatons and then they can be assgned to tasks and scheduled wth the gven set of processors as n the case of DSVF flter. Basc dea of MPC control s dsplayed n Fgure 5.4. Fgure MPC control Reference sgnal and dscrete states are brought to the block wth model-based predctve controller. The actual computed value of control sgnal s at the output of the controller, whch s connected wth the controllable nput (or more nputs) of the system. Functon of the MPC controller can be tested and optmzed va computer smulaton. If the strategy wth recedng horzon s used, sequence of control sgnal values s computed n every sample tme of the controlled system. The stuaton s the same as for DSVF flter smulaton; to update one sample of the output sgnal, a set of operatons has to be executed so the processor (or more processors) must have suffcent frequency to perform gven operatons n requred tme. In the real world cases, mathematcal operatons are assgned to processors n tme va some schedulng strategy. Therefore, the smulaton may take place n MPC control as well Applcaton of the Vsualzaton The vsualzaton serves as a tool to gan a better dea about realzaton of some problem n the frst place. In the schedulng area, vsualzaton can also serve as an optmzaton tool or as a verfcaton for already schedulng problems. Ths verfcaton s necessary e.g. for the producton schedulng where not only tme constrants has to be 32

49 Dploma Thess CTU n Prague, 2008 satsfed but also constrants caused by the adjustment n the space are mportant. These restrctons due to localzaton and movement of the materal or machnes are hard to cover up by the mathematcal descrpton and therefore hard to be taken nto account for the schedulng process. A representatve example s the host schedulng problem descrbed n Secton 2.2. The goal of the schedulng algorthm for the host schedulng problem s to schedule moves of one or more hosts wth materal, whch s processed n several tanks wth lqud. Fgure 5.5 dsplays the basc dea of the host schedulng problem. In some cases, the load and the unload staton can be merged nto one load/unload staton. Fgure Host schedulng problem Vsualzaton of the host schedulng problem can prove feasblty of the gven tme schedule or detect new restrctons that must be nvolved n the schedulng algorthm. Space restrctons are mportant especally n stuaton wth more hosts when ther trajectores may ntersect. Another area for vsualzaton of producton actvtes are process flows. For the purpose of producton cost reducton and productvty of work growth, amount of actvtes that do not yeld economc proft has to be mnmzed. As a consequence, producton operatons that do not add a value to the product have to be elmnated or suppressed at least. Materal transfers, machne setup tmes, tool handovers and cleanng are examples of actvtes to be avoded. Process flow s two or three-dmensonal transcrpt of the producton plan. It s formed by lnes and curves that connect machnes, buffers and other statons n the graphcal approach of the producton plan. Dagram of the movement n the space s very mportant for projectng the allocaton of machnes, buffers, tool repostores and other statons. Ths type of the vsualzaton plays the role durng the plannng of producton plans and t can be also used for representaton of the already scheduled soluton. An example of the 33

50 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek process flow s depcted n Fgure 5.6. Each object wth label M represents one machne for processng of the materal and each object wth label B represents one buffer for temporary storage of the materal. Sold curves stand for materal transfers, dashed curves represent moves of requred nstruments and dotted curves stand for garbage collecton. Fgure Example of process flows 34

51 Dploma Thess CTU n Prague, 2008 Chapter 6 6. Implementaton of VISIS As mentoned n Chapter 1, VISIS s ntended to be a tool for the smulaton and vsualzaton of the schedulng results. VISIS s mplemented n the Matlab envronment usng data structures and algorthms from TORSCHE Schedulng Toolbox for Matlab [7] and t wll be a part of ths toolbox n next release planned on October Users of VISIS can defne ther own project n the Vrtual Realty toolbox, standard part of Matlab, and bnd ths defnton wth Matlab commands. Both smulaton and vsualzaton are then realzed n Smulnk, whch s also a default part of the Matlab envronment. The VISIS mplementaton provdes several functons avalable for users and there are also several supplemental functons. In order to maxmum smplcty of usage, resultng Smulnk model s generated automatcally. Ths output model contans one masked subsystem representng the control system. In case of the vsualzaton, there s another block referencng to the predefned vrtual realty world. The mask of the control subsystem has nputs and outputs wth userdefned names and szes. The core of the control subsystem s the S-Functon block, whch contans man control functon. Ths functon realzes updatng of outputs accordng to the gven schedule and actual values of nputs. Ths control functon s also generated automatcally and all needed external data are created n Matlab workspace before the start of the smulaton. The S-Functon block has only one nput and output port as default so the n/out sgnals are ntegrated/dvded to reach user-defned number of nputs and outputs. Ths subsystem s then masked as one block wth approprate ports. The Smulnk model and code for the S-Functon block are both generated as text fles from the prepared templates. The control functon s called for each sample of Smulnk and the outputs are updated accordng to the schedule and actual Smulnk tme Implemented Functons All functons mplemented durng realzaton of the VISIS applcaton are standard Matlab m-fles. In ths subsecton, we present all of them wth descrpton of nput and output varables and command syntax. 35

52 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek adduserparam - functon that loads commands form the text fle and assgn them to tasks due to the format of the gven text fle. Commands from the text fle are stored n attrbute UserParam for each task. Some parameters can be also stored n attrbute TSUserParam of the taskset. Syntax: TSout = addcode( TS, fle) Where TS s a taskset object, fle s a strng wth the name of the text fle and TSout s output taskset wth assgned commands. In the text fle, each task has to be ntroduced by ts dentfer whch can be ts name (property Name of the task) or by specal character # and then ordnal number of a task n the taskset. Followng set of commands s assgned to an approprate task up to the keyword endparam. After ths keyword, new set of commands ntroduced by new dentfer of a task can be wrtten. Commands to be stored to attrbute TSUserParam of the taskset have to be ntroduced by the keyword UserParam.begn and termnated by the keyword endparam. Detaled utlzaton of ths functon s descrbed n Appendx A. All commentares (ntroduced by the character %) nsde the task defnton are coped to the S-Functon and all other are gnored. setports - functon that serves for settng the name and sze of ports of the control block n Smulnk. Syntax: ports = setports(varargn) Where ports s the output structure contanng nformaton about user-defned names and szes. Input of ths functon s alternatve and ts length depends on the amount of nputs and outputs that are needed for the concrete applcaton. Settng of nputs starts wth keyword Input and arbtrary amount of nputs can be lsted n the form (nput_name, sze_of_nput). Sze of nput represents length of the vector correspondng to an approprate port. The settng of outputs s the same, only wth the keyword Output now. An example of the functon call s n Appendx A agan. 36

53 Dploma Thess CTU n Prague, 2008 VRcontrol Syntax: - functon for settng of objects of vrtual realty and ther propertes that wll be controlled from Smulnk. Inputs of the vrtual realty block are automatcally created concernng ths defnton. VRn = VRcontrol(varargn) Where VRn s an output structure contanng nformaton about user-defned names and propertes of the vrtual realty (VR) objects. As an nput for ths functon, one or more pars of the exact VR object name and ts property can be defned n the form (object_name, object_property). Only numercal parameters of VR objects and strngs of text boxes can be controlled from the Smulnk model. taskset2smulnk - man functon of VISIS. Ths functon generates the Smulnk model and all other fles and data structures needed for the smulaton and vsualzaton. Syntax: taskset 2 smulnk(fle, TS, ports, VRn, stoptme, varargn) Where fle s a strng wth name of the vrtual realty fle that must be termnated by the suffx.wrl. If the VR s not used, arbtrary name of project can be set or empty parameter [] can be passed to the functon and all created data wll have default prefx project1. TS s the taskset object wth ncluded schedule and Matlab commands assgned to tasks. Structure ports s the parameter that can be obtaned by the setports functon and VRn s the structure resultng from the VRcontrol functon. Parameter stoptme serves to set duraton of the smulaton or vsualzaton n Smulnk. Some addtonal nformaton can be passed to the functon va arbtrary nput. User of VISIS can set the sample tme of the smulaton (default value s equal to one), perod of schedule repeatng and t s also possble to generate only the control functon nstead of the whole Smulnk model. Detaled descrpton of nput parameters s n the Appendx A. 37

54 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek The man functon taskset2smulnk handles arbtrary nputs, f any exsts, n the frst place. To be avalable for the resultng Smulnk model, four varables are assgned wth ther names n the workspace of Matlab - fle, perod, TS and sampletme. The code for the control S-Functon s then generated and f there s any structural error n the created code, a warnng s dsplayed. The last step of the taskset2smulnk s to generate the output Smulnk model and f there s no error, the model s opened. sfunctoncode - supplemental functon, whch generates code for the S-Functon regardng to the text assgned to tasks. The functon s called from the man functon taskset2smulnk and a new m-fle s saved to the current drectory of Matlab. Syntax: sfunctonc ode(fle, TS, ports, VRn, dspvr) Where fle contans the name of the project, TS s the taskset object wth assgned code, ports s the structure resultng from the setports functon and VRn s the structure resultng from the VRcontrol functon. Parameter dspvr s a bnary varable that determnes f the vrtual realty s used or not. Functon generates code for the S-Functon as a strng that s nserted to a new text fle afterwards. Ths fle s saved as S_projetName.m where projetname s the name defned by user or project1 as default. For the generaton of the S-Functon code, predefned template fle s used, one for the case wth vrtual realty (fle SFunctonBase.m) and one for the case wthout VR (fle SFunctonBase2.m). Some text from the template s coped and the rest s added accordng to sze of taskset and commands assgned to tasks. parsetask - supplemental functon that loads data from the task attrbute UserParam and transforms them to the form for the S-Functon code. Syntax: parsetask( T, ndex1, ndex2 ) Where T s a task object, whch contans code to be transformed, ndex1 s the ndex of start tme of task T n the vector of dscrete states of the S-Functon (wll be descrbed later) and 38

55 Dploma Thess CTU n Prague, 2008 ndex2 s the ndex of completon tme of task T n the vector of states. Ths functon s called from the sfunctoncode for each task n the taskset. smulnkmodel - supplemental functon servng to generate the output Smulnk model. The model s generated as a text fle from the predefned template fles. The functon s called from the man functon taskset2smulnk and a new Smulnk model fle s saved to the current drectory of Matlab. Syntax: smuln kmodel(fle, stoptme, sampletme, ports, VRn, dspvr) All nput parameters have the same meanng as for the prevous functons. The functon generates system wth one S-Functon block, one mux (multplexer) block and one demux (demultplexer) block. Amount of ports for both the mux and demux blocks are adjusted wth regard to count of user-defned nputs and outputs. The S-Functon block n Smulnk has just one nput and one output port, so the sgnals have to be ntegrated/dvded to satsfy gven demands for count and names of the nputs/outputs. The whole system s then masked as one subsystem wth approprate ports. If the vrtual realty s also defned, another block (VRsnk), referencng to the gven VR fle, s also added to the model. Smulnk model s generated as a strng and t s nserted to a new text fle afterwards. Ths fle s saved as projetname.mdl where projetname s the name defned by the user or project1 as default. getvrpar - functon that allows to obtan any numercal or strng value of the objects from the vrtual realty represented by the VRsnk block n Smulnk. Syntax: value = getvrpar(fle, object, var) Where fle contans the name of the project, object s a strng wth the exact name of the object n the VR fle, var s the name of the parameter to acqure the value from and value s the value of requred parameter. The getvrpar functon serves manly to get values of parameters that are not drectly controlled from the Smulnk model. Parameters, whch are 39

56 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek controlled va nputs of VR block n Smulnk are avalable n the S-Functon under the same name. sn - supplemental functon that s used n the S-Functon code to recognze f the followng set of commands s to be executed n the current tme. Syntax: status = sn(start, stop, perod, t) Where start and stop are the numercal values representng start tme and completon tme of a task n the schedule, perod s the perod of the schedule and t s the current tme n Smulnk. Output bnary varable status s equal to one f and only f a task determned by ts start tme and completon tme s to be executed n current tme respectng the perod of the schedule. Functons adduserparam, setports, VRcontrol, taskset2smulnk and getvrpar are ntended for drect work wth VISIS whereas sfunctoncode, parsetask, smulnkmodel and sn are supplemental functons not mentoned to be used by users of VISIS Smulnk Model Descrpton Both the smulaton and vsualzaton of schedulng results are realzed n Smulnk, whch s a part of Matlab ntended for wde spectrum of smulatons. The Smulnk model s generated automatcally and t contans one control block wth ncluded S-Functon block and n the case of vsualzaton, there s also one VR snk block referrng to the gven fle wth the vrtual world defnton. The S-Functon block s ntended for executng of Matlab commands, wrtten n standard Matlab m-fle, durng the smulaton n Smulnk. Inputs for the S- Functon block are the name of the assgned m-fle, names of varables n the workspace to be passed to the S-Functon and the last nput s defnton of addtonal modules. Dalog box for the S-Functon block s depcted n Fgure 6.1. The S-Functon block tself has one nput port and one output port. Length of the vector that wll be accepted n the nput of the block and the vector length of the output are defned n the assgned m-fle. Input vector s avalable under the name u n the S-Functon and more, actual Smulnk tme s stored n varable t, 40

57 Dploma Thess CTU n Prague, 2008 vector of nternal states s n the varable x and flag s automatcally updated varable, whch determnes actual step n the S-Functon call. Fgure S-Functon dalog box Executon of the S-Functon s separated nto several steps for each functon call n dependence on the type of nternal states and requred sample tme of operatons. Decson about next routne to be executed s made on bass of the actual value of the flag varable. If the S-Functon block s called for the frst tme, ntalzaton routne s executed. Durng ths routne, number of contnuous and dscrete states, number of nputs and outputs, sample tme of the functon and ntal values of the nternal states are defned. After the ntalzaton, subfuncton for updatng output values s called. If there are any contnuous states (not n case of VISIS mplementaton), next step of the functon s to compute ther dervates. Dscrete states are updated after the contnuous dervates computaton and f the varable sample tme of the S-functon s defned, next tme moment for the S-Functon block callng s computed. After the last step of the smulaton, termnaton routne s executed. All these subfunctons except the ntalzaton and termnaton are executed durng each call of the S-Functon. VISIS mplementaton nvolves only dscrete states and the sample tme s fxed, so the steps for dervates and next callng tme computaton are skpped. In the ntalzaton, number of dscrete states, nputs and outputs are automatcally computed n regard to gven taskset and user-defned nput and output ports. Start tmes, processng tmes and processors specfcaton are read from the taskset and saved to the vector of nternal states. Values of the 41

58 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek controlled nputs of vrtual realty are assgned to approprate output varables and saved to nternal states too. User-defned code assgned to tasks s ncluded n the subfuncton for updatng dscrete states. Current dscrete step s computed from the actual tme of the Smulnk smulaton and current values of controlled nputs of the vrtual realty block are assgned to approprate varables va functon getvrpar. Functon sn s used to determne whch task s executed n current step of the smulaton n regard to the gven tme schedule. Tasks can be dvded nto more parts (see Appendx A), so the decson s made for each part of each task separately. At the end of dscrete states updatng, all computed values are stored to predefned postons n the vector of nternal states, so all these values wll be avalable n the next step of the smulaton. The last subfuncton, called n one sample of the S-Functon block, updates outputs. An approprate part of the vector of nternal states s coped to the output vector n ths subfuncton. The S-Functon block s connected wth requested number of nputs and outputs and the whole system s then masked as one subsystem n Smulnk. Ths mask s called projectname_subsystem where projectname s a name specfed by user of VISIS and the nputs and outputs of ths masked subsystem correspond wth user defnton. Example of the generated subsystem s dsplayed n Fgure 6.2 and ts mask s dsplayed n Fgure 6.3. Fgure Subsystem wth S-Functon block 42

59 Dploma Thess CTU n Prague, 2008 Fgure Mask of the subsystem As we can see n Fgure 6.2, any unused port s termnated by the ground to avod warnngs and errors caused by not connected ports. All Smulnk objects are generated automatcally and ther szes, postons and number of ports are also set automatcally. If the vrtual realty s used, block referrng to specfed VR fle s also ncluded. Fgure 6.4 dsplays the whole Smulnk model, whch s a result of onehost_demo example. Fgure Smulnk model wth VR block 43

60 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek It s possble to add any other object to the generated Smulnk model and then start the smulaton. Vrtual realty world s automatcally opened n the case of vsualzaton Task Representaton Each task s determned by ts processng tme and start tme n the schedule. Completon tme of a task can be easly computed from these values. All operatons defned for a task are executed n regard to these numercal values. To allow as much precse problem representaton as possble, each task can be separated to more parts wth dfferent commands and dfferent duraton. Each part of a task s relatve to ts start tme n the schedule. There are three ways how to defne operatons for one task n tme (see Appendx A). Commands assgned to a task can be repeated for a defned number of samples or executed just once. In regard to Fgure 5.2 we can defne operaton of loadng data executed n the tme moment correspondng wth the start tme of a task, operaton of uploadng data n the moment correspondng to completon tme and any repeatng operaton executed for every sample of the Smulnk smulaton between these tme moments Vrtual Realty Toolbox Graphcal objects for the vsualzaton are created n VRedt (part of Vrtual Realty toolbox for Matlab). VRedt allows to defne basc geometrcal objects, text, background, textures and complex objects. VR toolbox lnks MATLAB and Smulnk wth vrtual realty graphcs, enablng MATLAB or Smulnk to control the poston, rotaton, dmensons, etc. of the 3-D mages defned n the vrtual realty envronment. To be controllable, the object n vrtual realty must have unque name. Ths dentfer has to be chosen as an nput of the VR block n Smulnk together wth the property that wll be controlled. In each sample of the Smulnk smulaton, propertes of the vrtual realty world are updated accordng to the vector values at the nput of VR block. VISIS generates the whole Smulnk model so the nput ports for VR block are defned automatcally. To change strng value n the vrtual realty world, functon setfeld can be used. Parameters of ths functon are the vrtual realty node, ts property and new requested value. Example of settng the strng value of the node Tank1 n the VR fle onehost.wrl: 44

61 Dploma Thess CTU n Prague, 2008 w=vrworld('onehost.wrl'); open(w) node = vrnode(w,'tank1'); setfeld(node,'strng','busy') close(w) 45

62 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Chapter 7 7. Case Studes In ths secton, performance measures for the new algorthm for the optonal schedulng problem are presented. Random generator of nstances for the problem wth alternatve process plans was realzed created and the algorthm was tested usng these data. Furthermore, four examples for the vsualzaton and one for the smulaton wth VISIS were created to show capabltes of VISIS and three of them are descrbed n ths secton Performance Measures for the Optonal Schedulng Problem There are three algorthms proposed for the optonal schedulng problem; one for the problem wth one processor, one for nfnty amount of dentcal processors and one for the case wth dedcated processors. For performance measures, algorthm for the problem wth one processor s used. Fgure 7.1 dsplays mean CPU tme used to solve the problem n dependence on number of transtons n a Petr net. All dsplayed results are average from 100 measurements wth randomly-generated nstances. These nstances consst of Petr net ncdence matrx W and vector of processng tme p. The ncdence matrx W results from a Petr net that represents nstance of the problem wth alternatve process plans. The mplemented generator of nstances creates one source place wth number of output transtons unformly generated from the vector v 1 = [ 2, 2, 2, 2, 2, 3, 3, 3, 4, 4]. Number of places s then unformly generated from the vector v 2 = [ 1,1,1,1,1, 2, 2, 2, 2, 3] and each transton s randomly connected wth one output place. For each place, number of output transtons s unformly generated from vector v 1 and these steps are repeated untl the specfed amount of transtons (tasks) s acqured. At the end, all transtons wthout output place are connected to the snk place. Each processng tme s generated unformly from the nterval <1;10>. The measurement was performed for the amount of tasks from 5 to 28. Tme complexty of the presented ILP model s exponental. 46

63 Dploma Thess CTU n Prague, 2008 Mean CPU tme [s] Number of tasks Fgure Mean solvng tme Fgure 7.2 dsplays dependence of memory sze used for solvng on the amount of tasks. Solvng demands for memory are very low n comparson wth the tme complexty. 3 Mean memory used [MB] Number of tasks Fgure Mean memory used Number of varables and constrants n dependence on sze of nput problem s dsplayed n Table 7.1. Varables bnary 2 n + n m + n nteger n Constrants m n ( m + n) Table Number of varables and constrants Where n s amount of PN transtons and m s amount of PN places. 47

64 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Fgure 7.3 shows for a gven amount of tme, how many nstances have been solved. Measurement was realzed for three quanttes of tasks wth one hundred nstances for each quantty. 100 Solved nstances [%] n=20 n=25 n= t [s] Fgure Rato of the solved nstances n tme All measurements was realzed on computer wth 1 GB operatonal memory and processor dual-core 1.6 GHz Smulaton wth VISIS Smulaton of Dgtal State Varable Flter (DSVF), descrbed n Chapter 5, was mplemented n Smulnk usng VISIS applcaton. Ths flter s formed by the set of equatons wth elementary arthmetc operatons. Each operaton s assgned to one task and n the text fle, t has the followng form: #1 n 0 x(8) = * x(7); n 3 x(1) = x(8); endparam Ths part of scrpt denotes that data for the operaton are loaded n the begnnng of frst task of the taskset n the schedule and the resultng output s avalable after three samples of task executng. All other operatons are assgned to tasks smlarly. Input for the flterng s 48

65 Dploma Thess CTU n Prague, 2008 stored n varable I and the output varable s named L n the commands for tasks. All other varables are stored n the vector of nternal states. Matlab commands for the defnton of smulaton wth VISIS are wrtten below. %Defne taskset and add code for tasks TS = taskset([ ]); TS = adduserparam(ts,'dsvf.txt'); %Defne perod of tasks perod = 11; %Set the schedule starts = [ ]; add_schedule(ts,'dsvf',starts,ts.proctme) %Defne parameters for the smulaton n Smulnk stop = 1; sample = 1/220000; %Defne nputs and outputs for the S-Functon block ports = setports('input','i',1,'output','l',1); %Call man functon taskset2smulnk('dsvf',ts,ports,[],stop,'perod',perod,'sample',sample); The Smulnk model resultng from ths smple defnton contans one masked subsystem wth nput I and output L. Pulse generator wth unt ampltude and perod 0.1s and the Scope unt are added to the model and the smulaton s ready to start, see Fgure 7.4. Fgure Smulnk model of DSVF 49

66 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek The result of the smulaton s the same as n case wth TrueTme tool, descrbed n Chapter 5. The nput and output sgnal of the smulated flter s dsplayed n Fgure 7.5; purple lne s the sgnal from the pulse generator and the yellow one s the fltered sgnal. Fgure DSVF smulaton sgnals 7.3. Vsualzaton wth VISIS Technque for the vsualzaton s very smlar to the case of the smulaton wth only one dfference durng defnton. Ports for the VR block n the Smulnk model have to be defned and the vrtual realty world has to be created ndeed. The frst realzed example s the vsualzaton of the host schedulng problem, descrbed n Chapter 5. More precsely, two vsualzatons for the host schedulng problem were created, frst one wth one host carryng materal and the second one wth two hosts. The materal has to be processed n three tanks wth lqud n both cases. For the stuaton wth only one host, load and unload statons are merged nto one place. Graphcal appearance was defned n the VRedt envronment and the project s desgned as a 2D vsualzaton. Controllable propertes are postons of the materal (represented by square objects wth dfferent textures), horzontal postons of the hosts, lengths of host arms and vertcal postons of host wrsts. Moreover, strng values representng the amount of watng and fnalzed materal are beng changed usng functon setfeld. The Smulnk model for the case wth one host s dsplayed n Fgure 6.4 and the ntal state of vrtual realty s n Fgure

67 Dploma Thess CTU n Prague, 2008 Fgure Vrtual realty for the Host schedulng Four tasks are needed to represent moves of the host wth the materal. The schedule wth these tasks s repeated perodcally and the commands to perform the moves are executed n regard to start tme of an approprate task n the schedule. Progress of the vsualzaton s slghtly demonstrated n Fgure 7.7. Fgure Progress of the vsualzaton Second example of the vsualzaton wth VISIS s the workshop for producton of small lamps. How to modfy poston, sze, rotaton and also color of objects s shown n ths example. One frame of the vsualzaton s dsplayed n Fgure

68 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Fgure Vsualzaton of the workshop The last realzed vsualzaton wth VISIS s the motvaton example to show mportance of the schedulng. Let us magne a rver and four solders standng on the shore n the nght. They have only one flash-lamp and to get on the other sde of the rver, at most two solders can walk together and they must have the flash-lght. Each solder can walk through the rver wth dfferent speed and movement of two solders s made wth the speed of the slower one. The goal s to transport all solders on the adverse sde of the rver n the shortest possble tme. Movements of solders are represented by tasks wth processng tmes equal to tme needed to cross the rver and two tme schedules are created, one wth the correct progress of solders transfers and one wth the ncorrect progress. Dfference n acqured tmes can show the purpose of the tme schedulng. One frame of the realzed vsualzaton for ths problem s shown n Fgure 7.9. Tmes needed for transports of solders are dsplayed on the left sde and actual tme value s dsplayed n the upper left corner. 52

69 Dploma Thess CTU n Prague, 2008 Fgure Motvaton example 7.4. Profler Results In ths subsecton, we wll show some results acqured from the Matlab profler, whch supports measurement of executon tme of functons. Profler returns nformaton about tme spent by each functon that s called durng the selected command executon. Number of functon calls and detaled lst of called subfunctons are also avalable. For each created example, presented n the prevous text, executon tmes of VISIS functons are measured and the results are stated n Table 7.2. Functon / Example Flter Host Workshop Solders adduserparam 0,29 s 0,34 s 0,49 s 0,34 s taskset2smulnk 1,72 s 2,46 s 2,78 s 4,22 s sftunctoncode 0,48 s 0,41 s 0,71 s 0,36 s smulnkmodel 0,53 s 0,55 s 0,62 s 0,52 s number of tasks Table Executon tmes To generate the Smulnk model and the ncluded S-Functon, t takes only a few seconds and t does not ncrease rapdly n dependence on the amount of tasks or the length of the gven text fle. Most of the executon tme of the man functon taskset2smulnk s spent 53

70 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek by openng of the generated Smulnk model and crtcal s the graphcal complexty of the vrtual realty fle. The Smulnk model fle and the S-Functon code are generated as strngs and then ncluded nto the new fle, so the tme complexty of generaton n dependence on amount of tasks and text fle length s lnear. Dstrbuton of tme between functons durng the smulaton of the Dgtal State Varable Flter was probed. The S-Functon s dvded nto several steps (see Chapter 6) and most of tme (17 from total 26 seconds) of ts executon s spent by the subfuncton for updatng dscrete states. Ths s the expected fact, because all commands specfed by user of VISIS are executed durng ths part of S-Functon. Tme needed to update dscrete states s dstrbuted unformly through the functon so there s no bottle-neck, whch would halt the smulaton. Smulaton by VISIS needs approxmately 80% of tme n comparson wth the same example realzed usng TrueTme lbrary. In addton, tme needed for one second of smulaton wth samples per second s approxmately 32 seconds n TrueTme and 26 seconds n VISIS. 54

71 Dploma Thess CTU n Prague, 2008 Chapter 8 8. Conclusons Ths work presents results acheved n two parts of schedulng area. New algorthm for the optonal schedulng problem has been proposed and VISIS, an applcaton for the vsualzaton and smulaton n schedulng, has been realzed n the Matlab envronment. Both parts offers soluton for problems that are usable n the area of producton schedulng. Ths branch of the schedulng theory plays an mportant role nowadays and tme savngs durng the producton plannng and schedulng lead to less demandng manufacturng of products. For the mplementaton purpose of the new schedulng algorthm for the optonal schedulng problem based on Integer Lnear Programmng (ILP), termnology arsng from the related works has been establshed n the frst place. Orgnal representaton by the Petr nets formalsm s then proposed. Ths concept of nput data assgnment allows to defne the problem structure naturally and also later modfcaton of the structure s smple. The ILP model s desgned from the state-transtons matrces of gven Petr net. Proposed soluton based on ILP soluton was tested on random-generated data and the results show that the algorthm s sutable to solve optonal schedulng problems wth up to 30 tasks n wthn a few mnutes wth low memory requrements (only about 5 MB for the bggest tested nstances). The algorthm s capable to solve the problems defned as a Petr net wth specfed propertes. Processng tme, release tme and deadlne can be assgned to each task. Proposed nteger lnear programmng model can be easly extended, new task parameters can be added and the optmalty crteron modfed. Establshed termnology wll be a base for the followng research n the area of the optonal schedulng. New optmal algorthms can be proposed and some polynomal heurstc can arse from the propertes of Petr nets. Moreover, utlzaton of Petr nets offers a possblty to use some of many exstng methods for ther analyss and smplfcatons. VISIS has two areas of use: n dscrete smulaton (e.g. n dgtal sgnal processng) and n the vsualzaton of scheduled problems. It s planed as an extenson of future verson of TORSCHE Schedulng Toolbox for Matlab. The applcaton can be used for presentatons, educatonal purposes or as an optmzaton tool and whenever clear presentaton of results s 55

72 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek needed. Durng the mplementaton of VISIS, several functons for work wth ths tool and several supplemental functons were realzed. Some demonstratve examples was created to show the capabltes of VISIS. Vsualzaton wth VISIS supports user-defned appearance of the vrtual realty and t allows to bnd ths defnton wth arbtrary Matlab commands. Up to our knowledge, there s no such a tool provdng vsualzaton of scheduled problems n that range. The smulaton n schedulng can serve as a fast feedback for results of schedulng process. Smulaton of dgtal state varable flter s faster than n TrueTme lbrary snce VISIS s optmzed for smulatons of tme schedules. The man advantage of VISIS s easer problem defnton and smple usage. The mplemented applcaton s desgned to maxmze the comfort and smplcty of utlzaton. 56

73 Dploma Thess CTU n Prague, 2008 References [1] R.BARTAK: Unary Resource Constrant wth Optonal Actvtes. In Lecture Notes n Computer Scence, Vol. 3258/2004, p [2] J.CH.BECK and M.S.FOX: Constrant-drected Technques for Schedulng Alternatve Actvtes. In Artfcal Intelgence, 2000, Vol. 121, p [3] A.WEINTRAUB, D.CORMIER, T.HODGSON, R.KING, J.WILSON and A.ZOZOM: Schedulng wth Alternatves: a Lnk Between Process Plannng and Schedulng. In IIE Transactons, 1999, Vol. 31, p [4] C.SAYGIN, F.F.CHEN, J.SINGH: Real-Tme Manpulaton of Alternatve Routengs n Flexble Manufacturng Systems: A Smulaton Study. In The Internatonal Journal of Advanced Manufacturng Technology, 2001, Vol. 18, p [5] E.VIN, P.LIT, A.DELCHAMBRE: A Multple-objectve Groupng Genetc Algorthm for the Cell Formaton Problem wth Alternatve Routngs. In Journal of Intellgent Manufacturng, 2005, Vol. 16, p [6] J.BLAZEWICZ et al: Schedulng n Computer and Manufacturng Systems. Sprnger, [7] P.ŠŮCHA, M.KUTIL, M.SOJKA and Z.HANZÁLEK: TORSCHE Schedulng Toolbox for Matlab. In IEEE Internatonal Symposum on Computer-Aded Control Systems Desgn, 2006, p [8] M.OHLIN, D.HENRIKSSON and A.CERVIN: TRUETIME 1.4 Reference Manual. Department of Automatc Control, Lund Unversty, [9] G.S.FISHMAN: Dscrete-Event Smulaton: Modelng, Programmng, and Analyss. Sprnger, [10] F.MANLIG and M.ŠRÁMEK: Řízení výrobních zakázek s podporou počítačové smulace. In journal Průmyslovés nženýrství, 2003, p [11] J.MISRA: Dstrbuted Dscrete-Event Smulaton. In ACM Computng Surveys (CSUR), 1986, Vol. 18, p [12] D.TOAL, T.COFFEY and P.SMITH : Expert Systems and Smulaton n Schedulng. CteSeer, [13] H.WWTTSTEIN, H.ZOLLER and G.LIEFLANDER: Vsualzaton of Process Schedulng. Unverstät Karlsruhe, Department of Computer Scence. 57

74 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek [14] A.SCHRIJVER: Theory of Lnear and Integer Programmng. Paperback, [15] T.MURATA: Petr Nets: Propertes, Analyss and Applcatons. In Proceedngs of the IEEE, 1989, Vol. 77, p [16] W.E.WILHELM: A Column-Generaton Approach for the Assembly System Desgn Problem wth Tool Changes. In Internatonal Journal of Flexble Manufacturng Systems, 1999, Vol. 11, p [17] R.HELIMANN: A Branch-and-bound Procedure for the Mult-mode Resource- Constraned Project Schedulng Problem wth Mnmum and Maxmum Tme Lags. In European Journal of Operatonal Research, 2003, Vol [18] G.TUNCEL, G.M.BAYHAN: Applcatons of Petr Nets n Producton Schedulng: a Revew. In The Internatonal Journal of Advanced Manufacturng Technology, 2007, Vol. 34, p [19] G.MEJIA, C.MONTOYA: A Petr Net Based Algorthm for Mnmzng Total Tardness n Flexble Manufacturng Systems. In Annals of Operatons Research, [20] R.L.GRAHAM, E.L.LAWLER, J.K.LENSTRA, A.H.G.RINNOOY KAN: Optmzaton and Approxmaton n Determnstc Sequencng and Schedulng Theory: a Survey. In Annals of Dscrete Mathematcs, 1979, Vol. 5, p [21] P.BAPTISTE, C.LE PAPE, W.NUIJTEN: Constrant-Based Schedulng - Applyng Constrant Programmng to Schedulng Problems. Sprnger, [22] J. W. HERRMANN: Handbook of producton schedulng. Sprnger, [23] H.KELLERER and V.A.STRUSEVICH: Schedulng Problems for Parallel Dedcated Machnes under Multple Resource Constrants. In Dscrete Appled Mathematcs, 2003, Vol.133, p [24] M.A.MANIER and CH.BLOCH A Clasfcaton for Host Schedulng Problems. In Internatonal Journal of Flexble Manufacturng systems, 2003, Vol. 15, p [25] J.LIU, Y.JIANG, Z.ZHOU: Cyclc Schedulng of a Sngle Host n Extended Electroplatng Lnes: a Comprehensve Integer Programmng Soluton. In IIE Transactons, 2002, Vol. 34, p [26] A.MAKHORIN: Modelng Language GNU MathProg. Documentaton for GNU Lnear Programmng Kt,

75 Dploma Thess CTU n Prague, 2008 [27] P.ŠŮCHA, Z. HANZÁLEK: Cyclc Schedulng of Tasks wth Unt Processng Tme on Dedcated Sets of Parallel Identcal Processors. In Proceedngs of the 3rd Multdscplnary Internatonal Conference on Schedulng: Theory and Applcaton, 2007, p [28] R.JOHNSON: Programmable State-Varable Flter Desgn for a Feedback Systems Web-Based Laboratory. Advanced Undergraduate Project Report, Massachusetts Insttute of Technology, [29] D.MATĚJÍČEK: Optmalzace Algortmů pro FPGA. Dploma Thess, CTU n Prague, [30] J.A.ROSSITER: Model-Based Predctve Control: A Practcal Approach. CRC Press, [31] E.F.CAMACHO, C.A.BORDONS: Model Predctve Control n the Process Industry. Sprnger,

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77 Dploma Thess CTU n Prague, 2008 Appendx A A. User Manual Ths project s realzed n the Matlab envronment [ more exactly n the verson Matlab R2006a. For older versons, there could be some ncompatblty, especally n case of creatng Smulnk model. Created functons can be called from command lne or from own m-fles. The result of whole project s the Smulnk model wth automatcally generated control functon and needed data structures. A.1. Smulaton by Smpler Substtuton of TrueTme Lbrary Frst necessary step after settng and schedulng the problem s to assgn Matlab operatons to tasks. These operatons wll be realzed n tme due to the fnal schedule. Ths code has to be stored n attrbute UserParam for each task n the taskset. E.g. for exstng task T 1 : code = 'n 0'; code = sprntf('%s\n%s\n',code,'x(1) = 5*x(1) 1;'); T1.UserParam = code; Ths way of addng code for tasks s qute dffcult and for larger amount of data also very mpractcal. Thus, a functon adduserparam s avalable. Ths functon adds code from a text fle to all tasks n the taskset together. Frst argument of ths functon s a taskset object and second argument s a strng wth the name of the text fle. Output object s a taskset wth assgned code. Example of use: TS = adduserparam(ts,'data.txt'); In the gven text fle, each task has to be ntroduced by ts name (T.Name) or by the specal character # and then ts ordnal number n the taskset, e.g. #3. Then t s possble to wrte down set of commands for an approprate task. Each set of commands has to be ended by the keyword endparam. If there are some commands, whch have to be executed at each sample tme of the smulaton, they have to be closed between keywords UserParam.begn I

78 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek and endparam. Any text outsde the bordered sets of commands, except the commentares, wll cause error. Commentares are not coped to the control functon. Example of text fle: UserParam.begn pause(0.1) endparam task1 y = 2*x^2; z = sn(x); endparam #2 x = x+1; endparam Next step for usng smulaton s to defne nputs and outputs of the Smulnk S- Functon block, whch calls control functon n each tme sample. Names of nputs and outputs wll be then accessble n task commands as varables. For ths defnton, functon setports s created. There are two keywords for ths functon: Input and Output. Frst of them ntroduces part for defnton of nputs of the S-Functon block and second one for outputs defnton. After each of them, next argument s the name of the nput/output and then ts sze (vector length). If only nputs (or outputs) are needed, only one keyword can be used. Output of ths functon s a structure that s one of the nput arguments for the man functon. Example how to create two nputs wth sze 1 and one output wth sze 2: ports = setports('input','w',1,'e',1,'output','control',2); Now s possble to call the man functon of the project taskset2smulnk. Ths functon wll generate Smulnk model and control functon and other data structures from the gven taskset wth tme schedule. Syntax of the functon call: taskset 2 smulnk(fle, TS, ports, VRn, stoptme, varargn) II

79 Dploma Thess CTU n Prague, 2008 Descrpton of parameters: fle - name of vrtual realty fle (has to be ended by postfx.wrl); all created functons and fles wll contan ths name. - f vrtual realty s not needed, arbtrary name of project can be set. - t s possble to set empty argument [] and project wll be named project1. TS - taskset object. ports - structure wth nformaton about nputs and outputs of the control block. VRn - structure wth nformaton about nputs of the vrtual realty block. - empty argument [] f vrtual realty s not used. stoptme - stop tme of the smulaton. varargn - addtonal nformaton; set n format: ( Property name, Property value). - Sample value of sample tme; default value s one. - Perod value of schedule perod; wthout repetton as default. - Smulnk strng wth nformaton about Smulnk model generaton. off f Smulnk model s not needed to be created. Example of the man functon call for the case wthout vrtual realty, name of project wll be dsvf, sample tme one second, stop tme 50 seconds, perod of schedule 10 seconds and we do not need to generate Smulnk model: taskset2smulnk('dsvf',ts,ports,[],50,'perod',10,'smulnk','off') A.2. Vsualzaton wth User-defned Vrtual Realty Technque for the vsualzaton of the schedulng results s the same as for the smulaton wth one addtonal step. It s necessary to defne nputs for the vrtual realty block n Smulnk. For ths purpose, functon VRcontrol s avalable. Input arguments are the pars of strngs, where frst one s the exact name of the object n the vrtual realty and second one s the property, whch we want to refer to. Output of ths functon s the structure wth gven nformaton. Example: VRn = VRcontrol('Arm1','translaton','ColorMachne1','dffuseColor'); III

80 Schedulng and Vsualzaton of Manufacturng Processes Roman Čapek Now, frst nput of the vrtual realty block wll refer to the object called Arm1 and t wll be possble to change the object property translaton va ths nput. If we want to control some nputs of the VR block by outputs of the control block, t s necessary to have these outputs/nputs at the same ports of blocks. For example, f we want to control the thrd nput of the VR block, an approprate output of the control block has to be also on the thrd poston. In the stuaton when more outputs from the control block than nputs to the VR block are needed, these outputs have to be defned after defnton of correspondng outputs/nputs. For example, f we want to connect frst two outputs of the control block wth the VR block and two other addtonal outputs are needed, the functon calls are followng: ports = setports('output','controlarm',3,'sgnalcolor',3,'u',1,'out',2); VRn = VRcontrol ('Arm1','translaton','ColorMachne1','dffuseColor'); It wll be possble to control translaton of the object Arm1 from the control block output controlarm. It s the same for the output sgnalcolor and the property dffusecolor of the object ColorMachne1. By callng man functon wth these structures together wth defned vrtual realty project, we get Smulnk model shown on Fgure. taskset2smulnk('my_poject.wrl',ts,ports,vrn,500); Fgure A.1 - Generated Smulnk model IV