research article the synchronized and integrated two-level ...research article the synchronized and...

Research ArticleThe Synchronized and Integrated Two-LevelLot Sizing and Scheduling ProblemEvaluating the Generalized Mathematical Model

Claudio F M Toledo1 Alf Kimms2 Paulo M Franccedila3 and Reinaldo Morabito4

1Department of Computer Systems Institute for Science Mathematics and Computing Universidade de Sao Paulo13566-590 Sao Carlos SP Brazil2Department of Technology and Operations Management Mercator School of Management University of Duisburg-Essen47057 Duisburg Germany3Department of Mathematics and Computer Science Universidade Estadual Paulista 19060-900 Presidente Prudente SP Brazil4Department of Production Engineering Universidade Federal de Sao Carlos 13565-905 Sao Carlos SP Brazil

Correspondence should be addressed to Reinaldo Morabito morabitoufscarbr

Received 4 December 2014 Revised 16 April 2015 Accepted 23 April 2015

Academic Editor Joaquim Joao Judice

Copyright copy 2015 Claudio F M Toledo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents the synchronized and integrated two-level lot sizing and scheduling problem (SITLSP) This problem isfound in beverage production foundry glass industry and electrofused grains where the production processes have usually twointerdependent levels with sequence-dependent setups in each level For instance in the first level of soft drink production rawmaterials are stored in tanks flowing to production lines in the second level The amount and the time the raw materials andproducts have to be stored and produced should be determined A synchronization problem occurs because the production in linesand the storage in tanks have to be compatible with each other throughout the time horizon The SITLSP and its mathematicalmodel are described in detail by this paper The lack of similar models in the literature has led us to also propose a set of instancesfor the SITLSP based on data provided by a soft drink company Thus a set of benchmark results for these problem instancesare established using an exact method available in an optimization package Moreover results for two relaxations proved that themodeling methodology could be useful in real-world applications

1 Introduction

The problem studied in this paper is motivated by a real situ-ation found in soft drink companies In the bottling industryproduction involves two interdependent levels with decisionsabout raw material storage in tanks and soft drink bottling inlines In the first level decisions regarding the amount andtime the raw materials have to be stored in each one of theavailable tanks should be made Similarly in the second levelthe lot size of each demanded item and its correspondingscheduling in each line should be determined A capacitatedlot sizing and scheduling problem has to be solved on eachone of these two levels (Figure 1)This problem covers variousissues of lot sizing and scheduling that have beendealtwith on

isolation only in the literature before The challenging aspectis the combination of all these issues in one interdependenttwo-level problem Capacitated lot sizing (and scheduling) isa topic of broad interest and has attracted many researchersWhile capacitated lot sizing (and scheduling) is an NP-hardoptimization problem [1 2] the problem of finding a feasiblesolution isNP-complete already [3] if setup times are present

One of the few optimal procedures for solving lot sizingand scheduling problems with sequence-dependent setuptimes can be found in Haase and Kimms [4] Kovacs et al(2009) presented a more compact model based on the pre-vious one proposed by Haase and Kimms [4] which allowssolving larger instances within a reasonable computationtime Luche et al [5] and Clark et al [6] presented mixed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 182781 18 pageshttpdxdoiorg1011552015182781

2 Mathematical Problems in Engineering

Tanks

Productionlines

Products

Figure 1 The two-level production process

integer programming (MIP) models for lot sizing and sched-uling in the presence of setup times for problems found inelectrofused grain and animal nutrition industries Concern-ing multilevel lot sizing problems several publications exist(eg [7ndash10]) The present paper deals with a multilevel lotsizing and scheduling problem which is difficult to tackleeven if the capacity is unlimited [11 12] and there are parallelmachines represented by production lines (Figure 1) Thework in de Matta and Guignard [13] deals with rolling prod-uction schedules for lot sizingwith parallelmachines A studyon sequence-dependent setup costs with parallel machinesis found in Kang et al [14] A lot sizing problem in a semi-conductor assembly with parallel machines was described byKuhn and Quadt [15] and Quadt and Kuhn [16] A GRASPheuristic with path-relinking intensification was applied byNascimento et al [17] to solve instances of the capacitated lotsizing problem with parallel machines

A R Clark and S J Clark [18] proposed a MIP formu-lation for a lot sizing and scheduling problem on parallelmachines with sequence-dependent setup times Their com-plex MIP formulation was solved using approximate modelsin a rolling horizon basis with a reduced number of binaryvariables per period Model reformulations were also pre-sented by Wolsey [19] to make standard software applicableClark [20] presented a mathematical model for the planningof a canning line in a drinks manufacturer He proposedapproaches using local search integrated with the solution ofapproximate MIP models Gupta and Magnusson [21] pre-sented a model for the capacitated lot sizing and schedulingproblem with sequence-dependent setup costs and nonzerosetup timesThe solver CPLEXwas used to find optimal solu-tions for small sized instances The authors also determinedlower bounds for large sized instances using row aggregationand relaxing the integrality of somemodel variables Almada-Lobo et al [22] proposed models for the same problem tack-led by Gupta and Magnusson [21] which were solved usingspecific-purpose heuristics and the CPLEX solver

A synchronization problem arises in a two-level problemreported by Almada-Lobo et al [23] and Toledo et al [24]Their focus is a glass container industry where the first leveldeals with the glass color produced in the furnaces and

the second level deals with the color of the containers that areproduced afterwards MIP formulations are tailored to tacklethis two-level and synchronized production planning prob-lem where metaheuristic approaches are proposed Baldo etal [25] studied another two-level problem in the breweryindustry preparing the liquids including fermentation andmaturation inside the fermentation tanks and bottling theliquids on the filling linesThe problemdiffers from soft drinkproblemsmainly due to the relatively long lead times requiredfor the fermentation and maturation processes and becausethe ready liquid can remain in the tanks for some time beforebeing bottled The surveys in Drexl and Kimms [26] Karimiet al [27] Jans and Degraeve [28 29] and Ramezanian et al[30] can be referred to for complete overviews in lot sizingproblems and industrial modeling for lot sizing problems

Awork that comes close to our case is the one described inMeyr [31] which is an extension of Meyr [32] Meyr [31] con-sidered a lot sizing and scheduling problemwith nonidenticalparallel machines inventory costs production costs andsequence-dependent setup times and costs However the softdrink industry problem in the present paper is more complexin the sense that it has two lot sizing and scheduling problemseach of them similar to Meyrrsquos problem one for lines andanother for tanks The problem solution should find simulta-neously the lot sizing and scheduling of rawmaterials in tanksand soft drinks in the bottling lines because there is interde-pendence between these two levels which is not present inMeyrrsquos approach Thus new variables and constraints able tosynchronize and integrate these two levels need to be addedThere are studies dealing with supply chain synchronizationor integration applied to decentralized manufacturers [33]However the synchronization aspect dealt with by the presentpaper is closer to the work of Tempelmeier and Buschkuhl[34]These authors introduced a synchronization problem ina multi-item multimachine lot sizing and scheduling prob-lemThis problem is found in the production system of auto-mobile suppliers There is a predetermined assignment ofproducts to machines where a common resource is necessaryfor setup processes on all machines

Bearing in mind the mentioned aspects the problemstudied by the present paper is called synchronized and inte-grated two-level lot sizing and scheduling problem (SITLSP)Someworks relatedwith the specific softdrink industrial situ-ation considered by the SITLSP have been published recentlyFor instance in Ferreira et al [35] a simplified formulationwas presented for the SITLSP where each bottling line hasa dedicated tank and each tank can be filled with all liquidflavors needed by this line Single-stage formulations werepresented in Ferreira et al [36] based on the general lot sizingand scheduling problem (GLSP) and on the asymmetric trav-elling salesman problem (ATSP) The solutions found usingthese formulations outperformed those found by solving theformulation described in Ferreira et al [35] Toledo et al [37]introduced a genetic algorithmmathematical programmingapproach using Ferreira et alrsquos [35] model which reachedcompetitive or better solutions for the real-world instancessolved in Ferreira et al [36] All these recent results assumethe hypotheses of dedicated tanks In our formulation thereare no dedicated tanks as it will be explained later

Mathematical Problems in Engineering 3

There are no existing modeling approaches to be directlyapplicable that include all the issues simultaneously consid-ered lot sizing scheduling setup (time and cost) parallelmachines among others Thus formulating a specific modelseems to be in order In Toledo et al [38] we have already pre-sented a related formulation of this paper (in Portuguese) in avery compact form but nowwe spend a lot of effort to explainall themodel constraints providing details about their mean-ings and interpretations An effective method to deal with allthe aspects considered by the SITLSP was first proposed byToledo et al [39] This method based on a multipopulationgenetic algorithm with individuals hierarchically structuredin trees was able to solve instances with several levels ofdifficulty based on data provided by a soft drink industry andthe solutions were compared with AMPLCPLEX solutionsreported on that paper

To sum up the main contributions of this paper are topresent in detail a mathematical formulation for the SITLP toevaluate its performance solving real-world problem instan-ces and to define sets of benchmark instances for the SITLSPAmathematicalmodel servesmany purposes as to provide anunequivocal definition of the problem better than a textualdescription can do The present paper combines ideas fromthe general lot sizing and scheduling problem (GLSP) withthe continuous setup lot sizing problem (CSLP) [26] todescribe the SITLSPThemodel is codednext and sets of small(artificially created) instances are solved using commercialsoftware packages to find optimum solutions which can beused for example as benchmark to evaluate heuristics pro-posed for the same problem Finallymedium-to-large instan-ces are created and solved applying some relaxation over theformulation when the optimal or even a feasible solution canbe found by the exact method

In the next section we describe the industrial problemin detail with some examples that will help us to explainmany specific problem issues In the section onMathematicalModels a MIP model is presented for the SITLSP In sectionon Computational Results we show some computationalresults obtained by solving the model with the modeling lan-guage AMPL in combination with the CPLEX solver Finallyin Conclusions we present concluding remarks and discussperspectives for future research

2 The Industrial Problem

Tomake the explanation easier the problem is posed as a softdrink industrial problem Taking this into account what it isreferred to as product in the problem is the combination ofthe soft drink flavor and the type of container Each of thesoft drinks is available in one or more bottle types such asglass bottles plastic bottles cans or bag-in-boxes of differentsizes In general only a subset of the bottling lines is capable ofproducing a certain product and products can be producedusing production lines in parallel Various products sharea common production line which cannot produce morethan one product at a time However whenever the productswitches a setup time (of up to several hours) is required toprepare the production line for the next product Production

lines can maintain the setup state When the production ofa product is preempted for a while and continued after someidle time then no setup is required Setup times are sequence-dependent meaning that the order in which products areproduced affects the required time to perform a setup

Let us take an example considering a time horizon of2 periods for a production process with 5 raw materials 6products 3 tanks and 3 lines Each time period has 5 timeunits of capacity Suppose the following

(i) Rm119860 (raw material 119860) produces 1198751 (product 1)(ii) Rm119861 produces 1198752 and 1198753(iii) Rm119862 produces 1198754(iv) Rm119863 produces 1198755(v) Rm119864 produces 1198756(vi) 1198711 (line 1) can bottle 1198751 1198752 and 1198753(vii) 1198712 can bottle 1198751 1198754 and 1198755(viii) 1198713 can bottle 1198751 and 1198756(ix) Tk1 (tank 1) can store Rm119860 Rm119861 and Rm119862(x) Tk2 can store Rm119860 Rm119861 Rm119862 and Rm119863(xi) Tk3 can store Rm119864

In the first time period the demands of products are 100units of 1198751 1198754 and 1198756 and 50 units of 1198755 In the second timeperiod the demands are 100 units of 1198752 and 1198753 and 50 unitsof 1198754 and 1198755 We are mentioning only relevant data for thisexample Figure 2 shows a possible schedule for a productionline level taking into account the line constraints as explainedbefore

The processing time for each product to be produced isshown in Figure 2 The assignments satisfied the set of prod-ucts allowed to be produced in each line and the demands ineach period There are also sequence-dependent setup timesfrom product 1198752 to 1198753 in 1198711 and from 1198754 to 1198755 in 1198712 Thisschedule only pays attention to the line constraints but it doesnot take into account the tank constraints discussed next

The raw material (ie the soft drink) that is bottled ina production line comes from a storage tank with limitedstorage capacity Different soft drinks cannot be put in a tanksimultaneouslyHence from time to time a tankmust be filledwith a particular soft drink which might be another drink orthe same as before Nevertheless whenever a tank is filled asignificant (sequence-dependent) setup time occurs to cleanand fill the tank even if the same soft drink is filled into thetank as before For technical reasons a tank can only be filledup when it is empty During the time of filling up a tanknothing can be pumped to a production line from that tankThere is no fixed assignment between tanks and productionlines Every tank can be connected to every production lineand a tank can feed more than one production line

In the previous example let us assume that all tanks have100 L (liters) of maximum storage capacity Suppose that 2 Lof Rm119860 is necessary to produce 1 u (unit) of 1198751 05 L of Rm119861

for 1 u of 1198752 and 1 u of 1198753 1 L of Rm119862 for 1198754 2 L of Rm119863 for1198755 and 1 L of Rm119864 for 1198756 Figure 3 shows a possible scheduleconsidering the tank constraints


0 1 2 3 4 5 6 7 8 9

Macro-period 1 Macro-period 2

Time10

L1L2L3

P1 P2 P3P4P5P4

P6P5

Figure 2 Schedule for the production line level

0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmA

RmD RmD

RmE

RmC

Tk1Tk2Tk3


Figure 3 Schedule for tank level

A total of 200 L of Rm119860 to produce 100 u of 1198751 isnecessaryTherefore tankTk1must be refilledwith Rm119860 andthe setup times are shown in Figure 3 The interdependencebetween the two levels of the problems becomes clear nowThe schedule in Figure 3 cannot be created without knowingthe line schedule in Figure 2 For instance it is necessaryto know when 100 u of 1198751 are scheduled on 1198711 to fill Tk1

with Rm119860 This also happens in Tk2 where it is necessaryto know when 1198754 is scheduled on 1198712 to fill Tk2 with Rm119863The schedules proposed in Figures 2 and 3 cannot be executedwithout repairs The same problem would happen if we hadscheduled the tank level first Figure 4 illustrates a possiblerepair that swapped product 1198754 for 1198755 from the schedules inFigures 2 and 3

Bottling is usually done to meet a given demand (perweek) with a production planning horizon of four weeks inparticular soft drink companies The decision maker has tofind out howmany units of what product should be producedwhen and on what production line To be able to do so thetanks must be filled appropriately The objective can includethe minimization of the total sum of setups inventory hold-ing and production costs Thus there is a multilevel (twolevel) lot sizing and scheduling problem with parallel proces-sors (tanks on the first level and lines on the second) capacityconstraints and sequence-dependent setup times and costsThe mathematical model proposed by this paper in the nextsection has to integrate these two lot sizing and schedulingproblems The production in lines and the storage in tanksmust be compatible with each other

3 A Model Formulation

This section introduces the mathematical model formulationfor the SITLSP First the basic ideas used by our formulationare described Next the objective function as well as each oneof the model constraints is explained Finally an example isused to show the type of information returned by the modelsolution

RmA RmB

RmC

RmE

RmA

RmD RmC


RmD

Tk1

Tk2

Tk3

L1

L2

L3

P1

P4

P2 P3P1

P5 P5 P4

P6

Figure 4 Occupation for tank and line slots

31 Basic Ideas The underlying idea for modeling our appli-cation combines issues from the general lot sizing and sched-uling problem (GLSP) and the so-called continuous setup lotsizing problem (CSLP) (see [26] for a comparison of thesemodels) A set of additional variables and constraints are alsoincluded in the model which can describe the specific issuesof the SITLSP

From theGLSPwe adopt the idea of defining a fixed num-ber (119878 for the production lines and 119878 for the tanks) of slots per(macro-) period so that at most one product or raw materialcan be scheduled per slotWhat needs to be determined is forwhat product or raw material respectively a particular slotshould be reserved in lines and tanks and what lot size (a lotof size zero is possible) should be effectively scheduled givena valid slot reservation Let us take the schedule shown inFigures 2 and 3 Suppose now 2macro-periods with 119878 = 119878 = 2

slots per period Figure 4 shows only the simultaneous slotreservation for lines and tanks in each period

The number of products in lines and raw materials intanks cannot be greater than the total number of slots (119878 = 119878 =

2) in each macro-period There is not a full slot occupationin Figure 4 For instance only one slot is effectively occupiedby raw materials and products respectively in tank Tk3 andline 1198713 during the first macro-period In the second macro-period there is no slot occupied in Tk3 and 1198713 Followingthe GLSP idea these two slots are reserved (1198756 in line 1198713 andRm119864 in tank Tk3) but nothing is produced

The use of slots in the model enables us to describe linesand tank occupation using variables and constraints indexedby slotsTherefore we can define decision variables like 119909119895119897119904 =

1 if slot 119904 in line 119897 can be used to produce product 119895 0 other-wise and119906119897119904 = 1 if a positive production amount is effectivelyproduced in slot 119904 in production line 119897 (0 otherwise) In thesame way we have decision variables like 119909119895119896119904 = 1 if slot 119904 canbe used to fill rawmaterial 119895 in tank 119896 0 otherwise and 119906119896119904 =1 if slot 119904 is used to fill raw material in tank 119896 0 otherwiseThese variables (and others) will define constraints for thecorrect line (or tank) occupation the reservation of only oneproduct (raw material) per slot and the effective production(storage) of products (raw materials) (see model constraints(2)ndash(10) and (30)ndash(39) in the next subsections)

In the SITLSP the slots scheduled on one level with theslots scheduled on the other level need to be coordinatedFor instance a production line level which must wait for asetup time on the tank level requires the tank setup time to


Table 1 Guideline for line and tank constraints

Line constraints Tank constraints Meaning(2) (30) Assignments not allowed for lines and tanks(3) (31) Assignments of products or raw materials to slots

(4) (16) and (17) (32)ndash(34) Link between the products or raw material assignments and the respective lot sizeproduced or stored

(7) (35) and (36) Products or raw materials changeover(8) and (9) (37) and (38) Changeover setup time(10) (39) Macro-period capacity(5) and (6) (64) and (65) Inventory balance(11)ndash(13) (41)ndash(43) Available capacity between endings of two consecutive slots(14) and (15) (44)ndash(47) Define only one beginning and only one ending for each slot(18)ndash(21) (23)ndash(24) (51)ndash(57) (59)ndash(61) Define the beginning and the ending for a slot effectively used or not(22) (58) Define an order for slot occupation(25) (28) and (29) (40) (48) (64) and (65) Synchronize when a slot of raw material is taken from tanks and produced by lines(26) and (27) mdash Use of the capacity of the first micro-period of each slotmdash (62) (63) and (65) Refilling of tanks

0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmC

RmE

RmA

RmD RmD RmC

Tk1Tk2Tk3L1

L2

L3

P 1 P 2 P 3P 1

P 5 P 4

P 6


Micro-periods

P 5P 4

Figure 5 Synchronization between slots in lines and tanks

be ready at least one small time unit before the productionto start in the line level For that reason we introduce smallmicro-periods in a CSLP-likemannerThemacro-periods aredivided into a fixed number of small micro-periods with thesame capacity This capacity can be partially used but thesame micro-period cannot be occupied by two slots Let usillustrate this idea in Figure 5 which synchronizes lines andtanks from Figure 4

Each macro-period was divided into 5 micro-periodswith the same length In the first period the productionof 1198751 1198754 and 1198756 must start after the initial setup time ofRm119860 Rm119862 and Rm119864 respectivelyThe rawmaterialmust bepreviously ready in the tank so that the production begins inlines The setup time needed to refill Tk1 with Rm119860 and Tk2with Rm119863 leads to an interruption of 1198751 and 1198755 productionin lines 1198711 and 1198712 respectively In the second period there isan idle time in1198712 after1198755 Line1198712 is ready to produce1198754 afterits setup time but the production has to wait for the Rm119862

setup time in Tk2Variables and constraints indexed by slots and micro-

periods will enable us to describe these situations Binary

variables 119909119861119897119904120591 and 119909

119864119897119904120591 indicate the micro-period 120591 when

the production of slot 119904 begins or when it ends in line 119897respectivelyThe analogous variables 119909

119861119896119904120591 and 119909

119864119896119904120591 are used to

indicate micro-period 120591 on slot 119904 of tank 119896 It is also necessaryto define linking variables 119902119896119895119897119904120591 which determine the amountof product 119895 produced in line 119897 in micro-period 120591 on slot 119904using raw material from tank 119896 These variables (and others)establish constraints which link production lines and tanksand synchronize raw material and product slots (see modelconstraints (25)ndash(27) (40) and (64) in the next subsections)

The model constraints are formally defined in the nextsubsections Table 1 has already separated them according tothe meaning making it easier to understand

32 Objective Function Suppose that we have 119869 raw mate-rials and 119869 products 119871 production lines and 119871 tanks Theplanning horizon is subdivided into119879macro-periods For themodeling reasons explained before we assume that at most 119878

slots per macro-period can be scheduled on each productionline and at most 119878 slots per macro-period and per tankcan be scheduled These slot values can be set to sufficientlylarge numbers so that this assumption is not restrictivefor practical purposes Furthermore for the same modelingreasons explained before we assume that each macro-periodis subdivided into119879

119898micro-periodsThenotation being usedin the model description is summarized in the appendix

As usual in lot sizing models the objective is to minimizethe total sum of setup costs inventory holding costs and pro-duction costsThe first three terms in expression (1) representsetup holding and production costs for production lines andthe next three the same costs for tanks Notice that a feasiblesolution which fulfills all the demand right in time withoutviolating all constraints to be describedmay not exist Hencewe allow that 119902

0119895 units of the demand for product 119895may not be

produced to ensure that the model can always be solved (lastterm in (1)) Of course a very high penalty 119872 is attached to


such shortages so that whenever there is a feasible solutionthat fulfills all the demands we would prefer this one

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119879

sum

119905=1ℎ119895119868119895119905 +

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1V119895119897119902119895119897119904

+

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119871

sum

119896=1

119879

sum

119905=1ℎ119895119868119895119896119905sdot119879119898

+

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1V119895119897119902119895119896119904 + 119872

119869

sum

119895=11199020119895

(1)

33 Production Line Usual Constraints First constraints forthe production line level are presented which are usuallyfound in lot sizing and scheduling problems

119909119895119897119904 = 0

119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(2)

119869

sum

119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)

119901119895119897119902119895119897119904 le 119862119909119895119897119904

119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(4)

1198681198951 = 1198681198950 + 1199020119895 +

119871

sum

119897=1

119878

sum

119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)

119868119895119905 = 119868119895119905minus1 +

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905

119895 = 1 119869 119905 = 2 119879

(6)

119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(7)

120590119897119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119897119911119894119895119897119904

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(8)

120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)

119905sdot119878

sum

119904=(119905minus1)119878+1(120590119897119904 +

119869

sum

119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905

119897 = 1 119871 119905 = 1 119879

(10)

A particular product can be produced on some lines butnot on all in general (2) and only one product 119895 is produced ineach slot 119904 of line 119897 (3)The available time permacro-period isan upper bound on the time for production (4)The inventorybalancing constraints (5) and (6) ensure that all demands aremet taking into account the dummy variable 119902

0119895 Constraint

(7) defines the setup times (119911119894119895119897119904 indicates this) based on slotreservations (119909119895119897119904) Thus it is possible to determine the setuptime 120590119897119904 to be included just in front of a certain slot (8) As aresult overlapping setup can happen and for the first slot ofa macro-period 119905 some setup time 120596119897119905 may be at the end ofthe previous macro-period (9) The capacity 119862 in constraint(10) has to be incremented by 120596119897119905 Thus the total sum of pro-duction and setup times in a certain macro-period must notexceed the macro-period capacity 119862 by constraint (10)

34 Production Line Time Slot Constraints The constraintsshowed next describe how to integrate the micro-period ideafrom CLSP and the slot idea from GLSPThemain point hereis to define specific constraints for the SITLSPwhich describewhen some slots in lines will be used throughout the timehorizon

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119905sdot119878120591 ge 120596119897119905+1

119897 = 1 119871 119905 = 1 119879 minus 1

(11)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904minus1120591 ge 120590119897119904

+

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(12)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897(119905minus1)119878+1120591 ge (119905 minus 1) 119862 + 120590119897(119905minus1)119878+1 minus 120596119897119905

+

119869

sum

119895=1119901119895119897119902119895119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

(13)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(14)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(15)

119869

sum

119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)

120576119906119897119904 le

119869

sum

119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(18)



0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)

10xEL1s410 minus 8xEL1s38 ge 120590L1s4 + pP3L1qP3L1

8xEL1s38 ge (2 minus 1) 5 + 120590L1s3 minus 120596L12 +

s4

pP2L1qP2L1s3

Figure 6 Keeping enough time between slots

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897(119905minus1)119878+1120591 minus ((119905 minus 1) 119879

119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)

The end of the last lot in some previous macro-period 119905

must leave sufficient time for the setup time 120596119897119905+1 (11) Thesame time slot control enables us to constrain that the finish-ing time of one slot minus the finishing time of the previousslot must be at least as large as the production time in thecorresponding slot plus the required setup time in advance(12)-(13) Figure 6 illustrates the situation described by theseconstraints with 119878 = 2 and 119904 = 1199041 1199042 1199043 1199044 120591 = 1 2 10119905 = 1 2 119862119898 = 1 and 119862 = 5

There is only one possible beginning and end for each slotin the time horizon (14)-(15) While 119909119895119897119904 indicates whether ornot a certain slot is reserved to produce a particular product119906119897119904 indicates whether or not that slot is indeed used to sched-ule a lot If the slot is not used then the lot size must be zero(16) otherwise at least a small amount 120576 must be produced(17) Time cannot be allocated for slots not effectively used(119906119897119904 = 0) so 119909

119861119897119904120591 = 119909

119864119897119904120591must occur for the samemicro-period

120591 (18)-(19) To avoid redundancy in the model whenever aslot is not used it should start as soon as the previous slotends (20)-(21) The possibly empty slots in a macro-periodwill be the last ones in that macro-period (22) The start andend of a slot are chosen in such a way that the start indicatesthe first micro-period in which the lot is produced if there isa real lot for that slot and the end indicates the last micro-period (23)-(24) Figure 7 illustrates the situation describedby these two constraints

The term 120576 sdot 119906119897119904 in constraint (23) and the termsum119869119895=1 119901119895119897119902119895119897119904 minus 119862

119898 in constraint (24) only matter for the casewhen the first and last micro-periods of a slot are the sameNotice that the processing time of 1198753 in Figure 7 is smallerthan the micro-period capacity in constraint (24) Howeverconstraint (23) ensures that a positive amount will be pro-duced because we must have 11990611987111198753 = 1 by constraint (16)

35 Production Line Synchronization Constraints Themodelmust describe how the raw materials in tanks are handled byproduction lines

119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4

120576 lowast uL1s3 + 8xEL1s38 minus 7xBL1s37 le pP2L1qP2L1s3


8xEL1s38 minus 7xBL1s37 ge pP2L1qP2L1s3

9xEL1s49 minus 9

minus 1

minus 1xBL1s49 ge pP3L1qP3L1s4

Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time

Figure 7 Time window for production

Macro-period 2 Constraint (28)

Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38

pP2L1qTk1P2L1s37 le xEL1s37



pP2L1qTk1P2L1s37 le xBL1s37


pP3L2qTk1P2L1s38 le xBL2s376 7 8 9 10

Time

Figure 8 Synchronization of line productions with tank storage

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

To link production lines and tanks we figure out whichamount of soft drink being bottled on a line comes fromwhich tank (25) In the first micro-period of a lot sometimes120575119897119904 are reserved for setup or idle time while the remainingtime is production time (26) and (27) Of course the softdrink taken from a tank must be taken during the time win-dow in which the soft drink is filled into bottles (28)-(29)Figure 8 illustrates the situation described by these con-straints with 119862

119898= 1 Let us suppose that Rm119860 is used to

produce 1198752 and 1198753Notice that the processing time of 1198753 takes from 120591 = 7 to

120591 = 8 This is considered by constraints (28) and (29)

36 Tanks Usual Constraints Now let us focus on usual lotsizing constraints for the tank level

119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time

From constraints (31)ndash(34)

Constraint (35)

Constraint (36)

Macro-period 2

xRmBTk2s4 = xRmBTk2s3 = uTk2s3 = 1

xRmDTk1s4 = xRmDTk1s3 = uTk1s4 = 1

zRmCRmDTk1s4 ge xRmDTk1s4 + xRmC Tk1s3 minus 2 + uTk1s4

xRmDTk1s4 minus xRmDTk1s3 le uTk1s4 997904rArr 1 minus 0 le 1 xRmBTk2s4 minus xRmB Tk2s3 le uTk2s4 997904rArr 1 minus 1 le 1

xRmCTk1s4 minus xRmC Tk1s3 le uTk1s4 997904rArr 0 minus 1 le 1

zRmCRmDTk1s4 ge 1 + 1 minus 2 + 1

zRmBRmB Tk2s4 ge xRmBTk2s4 + xRmBTk2s3 minus 2 + uTk2s4zRmBRmB Tk2s4 ge 1 + 1 minus 2 + 1

Figure 9 Raw material changeover

120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)

A particular rawmaterial can be stored in some tanks (30)and tanks can store one raw material at a time (31) The tankshave a maximum capacity (32) and nothing can be storedwhenever the tank slot is not effectively used (33) Howevera minimum lot size must be stored if a tank slot is used forsome rawmaterial (34)The setup state is not preserved in thetanks that is a tank must be setup again (cleaned and filled)even if the same soft drink has been in the tank just beforeCare must be taken when setups occur in the model only ifthe tank is really used because changing the reservation for atank does not necessarily mean that a setup takes place (35)-(36) Figure 9 illustrates these constraints

From the schedule in Figure 9 constraints (31)ndash(34)define 119909Rm119861sdotTk21199044 = 119909Rm119861Tk21199043 = 119906Tk21199043 = 1 and119909Rm119863Tk11199044 = 119909Rm119863Tk11199043 = 119906Tk11199044 = 1 Moreover the rawmaterials scheduled will satisfy constraints (35)-(36) As inthe production lines it is also possible to determine the tankslot setup time (37) and the setup time that may be at theend of the previous macro-period (38)-(39) Thus the setuptime in front of lot 119904 at the beginning of some period is thesetup time value minus the fraction of setup time spent in theprevious periodsThis value cannot exceed the macro-periodcapacity 119862

37 Tanks Time Slot Constraints The main point here is tokeep control when the tank slots are used throughout the time

horizon by production lines This will help to know when asetup time in tanks will interfere with some production lineor from which tanks the lines are taking the raw materials

119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

ge (119905 minus 1) 119862 + 120590119896(119905minus1)sdot119878+1 minus 120603119896119905

(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus ((119905 minus 1) 119879119898

+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)

119896 = 1 119871 119905 = 1 119879 minus 1 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 + 1120591 = (119905 minus 1)119879

119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898

Most of these constraints are similar to those presentedfor the production line level The amount that is filled intoa tank must be scheduled so that the micro-period in whichthe rawmaterial comes into the tank is known (40) Tank slotsmust be scheduled in such a way that the setup time that cor-responds to a slot fits into the timewindowbetween the end ofthe previous slot and the end of the slot under consideration(41)-(42) The setup for the first slot in a macro-period mayoverlap the macro-period border to the previous macro-period (43) This is the same idea presented by Figure 6 forlinesThefinishing time of a scheduled slotmust be unique asmust its starting time (44)ndash(47)The rawmaterial is availablefor bottling just when the tank setup is done (48) Thereforewe must have 119909

119864Tk211990437 = 119909

119864Tk211990449 = 1 in constraint (48) for

the example shown in Figure 10


Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49

Constraint (48) QTk2 = 100l

qRmCTk2s36 le 100 middot xTk2s37E


Figure 10 Tank refill before the end of the setup time

The time window in which a slot is scheduled must bepositive if and only if that slot is used to set up the tank(49)ndash(54) Redundancy can be avoided if empty slots arescheduled right after the end of previous slots (55)ndash(57)More redundancy can be avoided if we enforce that withina macro-period the unused slots are the last ones (58)

Without loss of generality we can assume that the timeneeded to set up a tank is an integer multiple of the micro-period Hence the beginning and end of a slot must bescheduled so that this window equals exactly the time neededto set the tank up (59)ndash(61) Constraint (61) adds up micro-periods 120591 = 1 119879

119898 seeking the beginning and end of thetank setup in slot 119904 = 1 Finally a tank can only be set up againif it is empty (62)-(63) Notice that a tank slot reservation(119909119861119896119904120591 = 1) does not mean that the tank should be previouslyempty (119868119895119896120591minus1 = 0)The tank slot also has to be effectively used(119906119896119904 = 1 and 119909

119861119896119904120591 = 1) in (62)-(63) Constraint (63) describes

this situation for the last macro-period 119879 where the first slotof the next macro-period is not taken into account

Finally notice that there are group of equations as forexample (45)ndash(47) (49)ndash(51) (52)ndash(57) (55)ndash(57) and (59)ndash(61) that are derived from the same equations respectively(45) (49) (52) (55) and (59) These derived equations arenecessary to deal with the first slot and sometimes with thelast slot in each macro-period

38 Tanks Synchronization Constraints The model mustdescribe how the products in production lines are handled bytanks

119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

minus 1What is in the tank at the end of a micro-period must be

what was in the tank at the beginning of that micro-period

0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4

Figure 11 Possible solution for SITLSP

0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2


Figure 12 Possible solution provided by the model

plus what comes into the tank in that micro-period minuswhat is taken from that tank for being bottled in productionlines (64) To coordinate the production lines and the tanksit is required that what is taken out of a tank must have beenfilled into the tank at least one micro-period ahead (65)

39 Example This subsection will present an example toillustrate what kind of information is returned by a modelsolution Let us start by the solution illustrated in Figure 5This solution is shown again in Figure 11 Figure 12 presentsthe same solution considering the model assumptions

According to the model the setup times of the tanks areconsidered as an integer multiple of the time of a micro-period For example the setup time of Rm119862 in Figure 11 goesfrom the entire 120591 = 1 (micro-period 1) until part of 120591 = 2 andfrom the entire 120591 = 8 until part of 120591 = 9 These setup times fitinto two completemicro-periods in Figure 12 (see constraints(59)ndash(61)) The same happens with the setup time of Rm119861


Table 2 Parameter values

Param Values Param Values Param Values Param Values119871 2 3 4 119871 2 3 ℎ119895 1 ($u) ℎ119895 1 ($u)119869 2 3 4 119869 1 2 ℎ119895 1 ($u) ℎ119895 1 ($u)119878 2 3 119878 2 V119895119897 1 ($u) V119895119896 1 ($u)119879 1 2 3 4 119879

119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u

Table 3 Parameter combination for each set of instances

Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342

(macro-period 2) The setup time of products in lines 1198711 and1198712 are positioned in front of the next slot as required byconstraints (8)ndash(13) and all the productions in the lines startafter the tank setup time is finished (constraints (25) (28)-(29) (40) and (48))

The satisfaction of the model constraints imposes thatbinary variables 119909119895119897119904 119909119895119896119904 119906119897119904 119906119896119904 119909

119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591

must be necessarily active (value 1) for lines tanks productsand raw materials in the slots and micro-periods shown inFigure 12These active variables provide enough informationabout what and when the products and raw materials arescheduled throughout the time horizon Moreover the con-tinuous variables 119902119895119897119904 119902119896119895119897119904120591 119902119895119896119904120591 119902119895119896119904 must be positive forlines tanks products and rawmaterials in the same slots andmicro-periods

For example based on Figure 12 the model solution haspositive variables 119902Rm119862Tk211990448 and 119902Rm119862Tk211990449 and returnsthe amount of Rm119862 filled in Tk2 during micro-periods 120591 =

8 and 120591 = 9 The total filled in this slot (1199044) is providedby 119902Rm119862Tk21199044 The model variable 119902Tk21198754119871210 returns theamount used by 1198754 in 1198712 which is taken from 119902Rm119862Tk21199044 Inthe sameway 119902Tk11198753119871111990449 and 119902Tk111987531198711119904410 have the lot sizesof 1198753 taken from slot 1199043 in variable 119902Rm119861Tk11199043 In this casethe total lot size 1198753 is provided by 119902119875311987111199044 All these variablesreturned by the model will provide enough information tobuild an integrated and synchronized schedule for lines andtanks

4 Computational Results forSmall-to-Moderate Size Instances

The main objectives in this section are to provide a set ofbenchmark results for the problem and to give an idea aboutthe complexity of the model resolution The lack of similarmodels in the literature has led us to create a set of instancesfor the SITLSP An exact method available in an optimiza-tion package was used as the first approach to solve theseinstances Table 2 shows the parameter values adopted

A combination of parameters 119871 119871 119869 and 119869 defines aset of instances These instances are considered of small-to-moderate sizes if compared to the industrial instances

found in softdrink companies Table 3 presents the parametervalues that define each combination 1198781ndash1198789 There are 9possible combinations for macro-period 119879 = 1 2 3 and4 For each one of these sets 10 replications are randomlygenerated resulting in 4 times 9 times 10 = 360 instances in totalThe other parameters of the model are randomly generatedfrom a uniform distribution in the interval [119886 119887] as shownin Table 4 Most of these parameters were defined followingsuggestions provided by the decision maker of a real-worldsoft drink companyThe setup costs are proportional to setuptimes with parameter 119891 set to 1000 This provides a suitabletradeoff between the different terms of the objective function

The computational tests ran in a core 2 Duo 266GHzand 195GB RAMThe instances were coded using the mod-eling language AMPL and solved by the branch amp cut exactalgorithm available in the solver CPLEX 110 The optimiza-tion package AMPLCPLEX ran over each instance once dur-ing the time limit of one hour Two kinds of problem solu-tions were returned the optimal solution or the best feasiblesolution achieved up to the time limit The following gapvalue was used

Gap() = 100(119885 minus 119885

119885

) (66)

where 119885 is the solution value and 119885 is the lower bound valuereturned by AMPLCPLEX Table 5 presents the computa-tional results for 119879 = 1 2 Column Opt has the number ofoptimal solutions found in each combination

Column Gap() shows the average gap of the final solu-tions returned by AMPLCPLEX from the lower boundfound The CPU values are the average time spent (in sec-onds) to return these solutions Table 6 shows the modelfigures for each combination In Table 5 notice that theAMPLCPLEX had no major problems to find solutions for119879 = 1 and119879 = 2The solverwas able to optimally solve all ins-tances for119879 = 1 (90 in 90) and found several optimal solutionfor 119879 = 2 (72 in 90) within the time limit The requiredaverage CPU running times increased from 119879 = 1 to 119879 = 2which is expected based on the number of constraints andvariables of the model for each instance (see Table 6)


Table 4 Parameter ranges

Par Ranges Meaning1205901015840119894119895119897 119880[05 1] Setup time (hours) from product 119894 to 119895 in line 119897

1205901015840

119894119895119896 119880[1 2] Setup time (hours) from raw material 119894 to 119895 in tank 119896

119904119894119895119897 119904119894119895119897 = 119891 sdot 1205901015840119894119895119897 Line setup cost

119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840

119894119895119896 Tank setup cost119901119895119897 119880[1000 2000] Processing time (unitshour) of product 119894 in line 119897

119903119895119894 119880[03 3] Liters of raw material 119894 used to produce one unit of product 119895

Table 5 AMPLCPLEX results for the set of instances with 119879 = 1 2

Comb 119879 = 1 119879 = 2

Opt Gap() CPU Opt Gap() CPU1198781 10 0 022 10 051 1901198782 10 0 048 9 085 585871198783 10 0 060 8 061 1271041198784 10 0 066 10 032 15491198785 10 0 146 8 152 1414511198786 10 0 1230 6 377 1519311198787 10 0 096 10 008 525461198788 10 0 437 6 079 2052721198789 10 0 783 5 138 182196

Table 6 Model figures for the set of instances with 119879 = 1 2

Comb119879 = 1 119879 = 2

Var Const Var ConstBin Cont Bin Cont

1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865

TheAMPLCPLEXwas also able to find optimal solutionsfor 119879 = 3 (43 in 90) and 119879 = 4 (26 in 90) within one hourbut this value decreased as shown in Table 7 when comparedwith those in Table 6 The complexity of the model proposedhere caused by the high number of binarycontinuousvariables and constraints as shown in Table 8 explains thehardness faced by the branch and cut algorithm to achieveoptimal solutions Nevertheless around 64 of the small-to-moderate size instances were optimally solved with 129nonoptimal (feasible) solutions returned The average gapfrom lower bounds Gap() has increased following thecomplexity of each set of instances However gap values arebelow 7 in all set of instances

These small-to-moderate size instances were already usedby Toledo et al [39] where the authors proposed a mul-tipopulation genetic algorithm to solve the SITLSP These

instances were solved and these solutions were helpful toadjust and evaluate the evolutionary method performanceThe insights provided by this evaluation were also useful toset the evolutionary method to solve real-world instances ofthe problem

5 Computational Results forModerate-to-Large Size Instances

The previous experiments with the AMPLCPLEX haveshown how difficult it can be to find proven optimal solutionsfor the SITLSP within one hour of execution time even forsmall-to-moderate size instances However they also showedthat it is possible to find reasonably good feasible solutionsunder this computational time In practical settings decisionmakers are usually more interested in obtaining approximate



Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4

Bin Var Cont Var Const Bin Var Cont Var Const1198781 320 803 863 430 1073 11631198782 574 2014 1835 574 2014 18271198783 598 2475 2075 598 2475 20751198784 480 1380 1303 645 1844 17521198785 642 2662 2071 861 3553 27641198786 669 3314 2350 897 4422 31401198787 558 1689 1536 749 2256 20391198788 768 3322 2466 1029 4433 33351198789 804 4190 2799 1077 5590 3735


Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812

solutions found in shorter times than optimal solutionswhichrequire much longer runtimes

For this reason instead of spending longer time lookingfor optimal solutions this section evaluates some alternativesto find reasonably good approximate solutions for moderate-to-large size instances For this another combination ofparameters 119871119871119869119869119879 is considered which defines a morecomplex and realistic set of instances namely 1198711ndash11987110Table 9 shows the corresponding parameter values

While the model parameters were changed to 119862 = 10 tu119879119898

= 10 119878 = 5 8 and 119878 = 3 4 the remaining parameterskept the same values of the ones shown in Tables 2 and 4For each one of the 10 possible combinations in Table 9 atotal of 10 replications was randomly generated resulting in100 instances Preliminary experiments with some of theseinstances showed that even feasible solutions are difficult tofind after executing AMPLCPLEX for one hour Thereforetwo relaxations of the model were investigated the first

Table 10 Relaxation approaches results

Comb 1198771 1198772

Opt Feas Penal Opt Feas Penal1198711 10 mdash mdash mdash 10 mdash1198712 10 mdash mdash mdash 9 11198713 9 1 mdash mdash 9 11198714 4 6 mdash mdash 2 81198715 8 2 mdash mdash 8 21198716 4 6 mdash mdash 3 71198717 6 4 mdash mdash 3 71198718 5 5 mdash mdash mdash 101198719 7 3 mdash mdash 7 311987110 mdash 5 5 mdash 4 6

relaxation 1198771 keeps as binary variable only the modelvariables 119906119897119904 and 119906119896119904 whereas the other binary variables areset continuous in the range [0 1] The second relaxation 1198772keeps 119906119897119904 and 119906119896119904 as binary as well as 119909119895119897119904 and 119909119895119896119904 A similaridea was used by Gupta ampMagnusson [21] to deal with a one-level capacitated lot sizing and scheduling problem

These two relaxation approaches were applied to solvethe set of instances 1198711ndash11987110 within one hour The resultsare shown in Table 10 where Opt is the number of optimalsolutions found by 1198771 and 1198772 respectively Feas is the num-ber of feasible solutions without penalties and Penal is thenumber of feasible solutions with penalties Remember that


0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)

Set of instancesL1 L2 L3 L4 L5 L6 L7 L8 L9

Optimal solutionsAll solutions

Figure 13 Average CPU time to find solutions

nonsatisfied demands are penalized in the objective functionof the model (see (1))

Indeed for moderate-to-large size instances approach1198771 succeeded in finding optimal solutions for a total of 63out of 100 instances Approach 1198772 returned only feasiblesolutions with some of them penalized A total of 55 out of100 solutions without penalties were found by 1198772

Figure 13 shows the average CPU time spent by 1198771 tosolve 10 instances on each set 1198711ndash1198719 This average time wascalculated for the whole set of instances as well as for onlythat subset of instances with optimal solutions found Allsolutions were returned in less than 3000 sec on average andoptimal solutions took less than 2000 sec Thus 1198771 and 1198772

approaches returned approximate solutions within a reason-able computational time for these sets of large size instances

6 Conclusions

A great deal of scientific work has been done for decadesto study lot sizing problems and combined lot sizing andscheduling problems respectively A whole bunch of stylizedmodel formulations has been published under all kinds ofassumptions to investigate this class of planning problemsIt is well understood that certain aspects make planningbecome a hard task not only in theory Among these issuesare capacity constraints setup times sequence dependenciesandmultilevel production processes just to name a fewManyresearchers including ourselves have focused on such stylizedmodels and spent a lot of effort on solving artificially createdinstances with tailor-made procedures knowing that evensmall changes of themodel would require the development ofa new solution procedure But very often it remains unclearwhether or not the stylized model represents a practicalproblemHence we feel that it is necessary tomove a bitmoretowards real applications and start to motivate the modelsbeing used stronger than before

An attempt to follow this paradigm is presented in thispaper We were in contact with a company that bottles very

well-known soft drinks and closely observed the main pro-duction process where different flavors are produced andfilled into bottles of different sizes The production planningproblem is a lot sizing and scheduling problem with severalthorny issues that have been discussed in the literature beforefor example capacity constraints setup times sequencedependencies and multilevel production processes But inaddition to the issues described in existing literature manyspecific aspects make this planning problem different fromwhat we know from the literature Hence we contributed tothe field by describing in detail what the practical problem isso that future researchers may use this problem instead of astylized one to motivate their research

Furthermore we succeeded in putting together a modelfor the practical problembymaking use of existing ideas fromthe literature that is we combined two stylized model for-mulations the GLSP and the CSLP to formulate a model forthe practical situation In this manner this paper contributesto the field because it proves a posteriori that the stylizedmodels studied before are indeed relevant if used in a properway which is not obvious As the focus of this paper is noton sophisticated solution procedures we used commercialsoftware to solve small and medium sized instances to givethe reader a practical grasp on the complexity of the problemA next step could be to work on model reformulations giventhat nowadays commercial software is quite strong if pro-vided with a strongmodel formulation Future workmay alsofocus on tailor-made algorithms of course We believe thatheuristics are most appropriate for such real-world problemsbut work on optimum solution procedures or lower boundsis relevant too because benchmark results will be needed

Appendix

Symbols for Parameters andDecision Variables

General Parameters

119879 number of macro-periods119862 capacity (in time units) within a macro-period119879119898 number of micro-periods per macro-period

119862119898 capacity (in time units) within a micro-period

(119862 = 119879119898

times 119862119898)

120576 a very small positive real-valued number119872 a very large number

Parameters for Linking Production Lines and Tanks

119903119895119894 amount of raw material 119895 to produce one unit ofproduct 119894

Decision Variables for Linking Production Lines and Tanks

119902119896119895119897119904120591 amount of product 119895 which is produced in line 119897

in micro-period 120591 and which belongs to lot 119904 that usesraw material from tank 119896


Parameters for the Production Lines

119869 number of products119871 number of parallel lines119871119895 set of lines on which product 119895 can be produced119878 upper bound on the number of lots per macro-period that is the number of slots in production lines119889119895119905 demand for product 119895 at the end of macro-period119905119904119894119895119897 sequence-dependent setup cost coefficient for asetup from product 119894 to 119895 in line 119897 (119904119895119895119897 = 0)ℎ119895 holding cost coefficient for having one unit ofproduct 119895 in inventory at the end of a macro-periodV119895119897 production cost coefficient for producing one unitof product 119895 in line 119897119901119895119897 production time needed to produce one unit ofproduct 119895 in line 119897 (assumption 119901119895119897 le 119862

119898)1199091198951198970 = 1 if line 119897 is set up for product 119895 at the beginningof the planning horizon1205901015840119894119895119897 setup time for setting line 119897 up from product 119894 to

product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)

1205961198971 setup time for the first slot in line 119897 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198971 gt 0 then values1199091198951198971 are already known and these variables should beset accordingly)1198681198950 initial inventory of product 119895

Decision Variables for the Production Lines

119911119894119895119897119904 if line 119897 is set up from product 119894 to 119895 at thebeginning of slot 119904 0 otherwise (119911119894119895119897119904 ge 0 is sufficient)119868119895119905 amount of product 119895 in inventory at the end ofmacro-period 119905119902119895119897119904 number of products 119895 being produced in line 119897 inslot 119904119909119895119897119904 = 1 if slot 119904 in line 119897 can be used to produceproduct 119895 0 otherwise119906119897119904 = 1 if a positive production amount is producedin slot 119904 in production line 119897 (0 otherwise)120590119897119904 setup time at line 119897 at the beginning of slot 119904120596119897119905 setup time for the first slot on production line 119897 inmacro-period 119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119897119904120591 = 1 if lot 119904 (which belongs to a single macro-

period 119905) in line 119897 ends in micro-period 120591119909119861119897119904120591 = 1 if lot 119904 (which belongs to a single macro-

period 119905) in line 119897 begins in micro-period 120591120575119897119904 time in the first micro-period of a lot which isreserved for setup time and idle time1199020119895 shortage of product 119895

Parameters for the Tanks

119869 number of raw materials119871 number of parallel tanks119871119895 set of tanks in which raw material 119895 can be stored

119878 upper bound on the number of lots per macro-period that is the number of slots in tanks119904119894119895119896 sequence-dependent setup cost coefficient for asetup from raw material 119894 to 119895 in tank 119897 119897 (119904119895119895119896 may bepositive)ℎ119895 holding cost coefficient for having one unit of rawmaterial 119895 in inventory at the end of a macro-periodV119895119896 production cost coefficient for filling one unit ofraw material 119895 in tank 1198961199091198951198960 = 1 if tank 119896 is set up for raw material 119895 at thebeginning of the planning horizon1205901015840119894119895119896 setup time for setting tank 119896up from rawmaterial

119894 to rawmaterial 119895 (1205901015840119894119895119896 le 119862) (wlog 1205901015840119894119895119896 is an integermultiple of 119862

119898)1205961198961 setup time for the first slot in tank 119896 in macro-period 1 that has already been performed beforemacro-period 1 starts (note if 1205961198961 gt 0 then values1199091198951198961 are already known and these variables should beset accordingly)1198681198951198960 initial inventory of raw material 119895 in tank 119896

119876119896 maximum amount to be filled in tank 119896119876119896 minimum amount to be filled in tank 119896

Decision Variables for the Tanks

119911119894119895119896119904 = 1 if tank 119896 is set up from raw material 119894 to 119895

at the beginning of slot 119904 0 otherwise (119911119894119895119896119904 ge 0 issufficient)119868119895119896120591 amount of raw material 119895 in tank 119896 at the end ofmicro-period 120591119902119895119896119904 amount of rawmaterial 119895 being filled in tank 119896 inslot 119904119902119895119896119904120591 amount of raw material 119895 being filled in tank 119896

in slot 119904 in micro-period 120591119909119895119896119904 = 1 if slot 119904 can be used to fill raw material 119895 intank 119896 0 otherwise119906119896119904 = 1 if slot 119904 is used to fill raw material in tank 1198960 otherwise120590119896119904 setup time that is the time for cleaning and fillingthe tank up at tank 119896 at the beginning of slot 119904120596119896119905 setup time that is the time for cleaning andfillingthe tank up for the first slot in tank 119896 inmacro-period119905 that is scheduled at the end of macro-period 119905 minus 1119909119864119896119904120591 = 1 if lot 119904 at tank 119896 ends in micro-period 120591

119909119861119896119904120591 = 1 if lot 119904 at tank 119896 begins in micro-period 120591


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their useful comments and suggestions They are alsoindebted to Companhia de Bebidas Ipiranga Brazil for itskind collaboration This research was partially supported byFAPESP andCNPq grants 4834742013-4 and 3129672014-4

References

[1] G R Bitran and H H Yanasse ldquoComputational complexity ofthe capacitated lot size problemrdquo Management Science vol 28no 10 pp 1174ndash1186 1982

[2] W-H Chen and J-M Thizy ldquoAnalysis of relaxations for themulti-item capacitated lot-sizing problemrdquo Annals of Opera-tions Research vol 26 no 1ndash4 pp 29ndash72 1990

[3] J Maes J O McClain and L N van Wassenhove ldquoMultilevelcapacitated lotsizing complexity and LP-based heuristicsrdquoEuro-pean Journal of Operational Research vol 53 no 2 pp 131ndash1481991

[4] K Haase and A Kimms ldquoLot sizing and scheduling withsequence-dependent setup costs and times and efficientrescheduling opportunitiesrdquo International Journal of ProductionEconomics vol 66 no 2 pp 159ndash169 2000

[5] J R D Luche R Morabito and V Pureza ldquoCombining processselection and lot sizing models for production schedulingof electrofused grainsrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 3 pp 421ndash443 2009

[6] A R Clark R Morabito and E A V Toso ldquoProduction setup-sequencing and lot-sizing at an animal nutrition plant throughATSP subtour elimination and patchingrdquo Journal of Schedulingvol 13 no 2 pp 111ndash121 2010

[7] R E Berretta P M Franca and V A Armentano ldquoMetaheuris-tic approaches for themultilevel resource-constrained lot-sizingproblem with setup and lead timesrdquo Asia-Pacific Journal ofOperational Research vol 22 no 2 pp 261ndash286 2005

[8] T Wu L Shi J Geunes and K Akartunali ldquoAn optimiza-tion framework for solving capacitated multi-level lot-sizingproblems with backloggingrdquo European Journal of OperationalResearch vol 214 no 2 pp 428ndash441 2011

[9] C F M Toledo R R R de Oliveira and P M Franca ldquoAhybrid multi-population genetic algorithm applied to solve themulti-level capacitated lot sizing problem with backloggingrdquoComputers and Operations Research vol 40 no 4 pp 910ndash9192013

[10] C Almeder D Klabjan R Traxler and B Almada-Lobo ldquoLeadtime considerations for the multi-level capacitated lot-sizingproblemrdquo European Journal of Operational Research vol 241no 3 pp 727ndash738 2015

[11] A Kimms ldquoDemand shufflemdasha method for multilevel propor-tional lot sizing and schedulingrdquo Naval Research Logistics vol44 no 4 pp 319ndash340 1997

[12] A Kimms Multi-Level Lot Sizing and Scheduling Methodsfor Capacitated Dynamic and Deterministic Models PhysicaHeidelberg Germany 1997

[13] R de Matta and M Guignard ldquoThe performance of rollingproduction schedules in a process industryrdquo IIE Transactionsvol 27 pp 564ndash573 1995

[14] S Kang K Malik and L J Thomas ldquoLotsizing and schedulingon parallel machines with sequence-dependent setup costsrdquoManagement Science vol 45 no 2 pp 273ndash289 1999

[15] H Kuhn and D Quadt ldquoLot sizing and scheduling in semi-conductor assemblymdasha hierarchical planning approachrdquo inProceedings of the International Conference on Modeling andAnalysis of Semiconductor Manufacturing (MASM rsquo02) G TMackulak J W Fowler and A Schomig Eds pp 211ndash216Tempe Ariz USA 2002

[16] DQuadt andHKuhn ldquoProduction planning in semiconductorassemblyrdquo Working Paper Catholic University of EichstattIngolstadt Germany 2003

[17] M C V Nascimento M G C Resende and F M B ToledoldquoGRASP heuristic with path-relinking for the multi-plantcapacitated lot sizing problemrdquo European Journal of OperationalResearch vol 200 no 3 pp 747ndash754 2010

[18] A R Clark and S J Clark ldquoRolling-horizon lot-sizing whenset-up times are sequence-dependentrdquo International Journal ofProduction Research vol 38 no 10 pp 2287ndash2307 2000

[19] L A Wolsey ldquoSolving multi-item lot-sizing problems with anMIP solver using classification and reformulationrdquo Manage-ment Science vol 48 no 12 pp 1587ndash1602 2002

[20] A RClark ldquoHybrid heuristics for planning lot setups and sizesrdquoComputers amp Industrial Engineering vol 45 no 4 pp 545ndash5622003

[21] D Gupta and T Magnusson ldquoThe capacitated lot-sizing andscheduling problem with sequence-dependent setup costs andsetup timesrdquo Computers amp Operations Research vol 32 no 4pp 727ndash747 2005

[22] B Almada-Lobo D KlabjanM A Carravilla and J F OliveiraldquoSingle machine multi-product capacitated lot sizing withsequence-dependent setupsrdquo International Journal of Produc-tion Research vol 45 no 20 pp 4873ndash4894 2007

[23] B Almada-Lobo J F Oliveira and M A Carravilla ldquoProduc-tion planning and scheduling in the glass container industry aVNS approachrdquo International Journal of Production Economicsvol 114 no 1 pp 363ndash375 2008

[24] C F M Toledo M Da Silva Arantes R R R De Oliveiraand B Almada-Lobo ldquoGlass container production schedulingthrough hybrid multi-population based evolutionary algo-rithmrdquo Applied Soft Computing Journal vol 13 no 3 pp 1352ndash1364 2013

[25] T A Baldo M O Santos B Almada-Lobo and R MorabitoldquoAn optimization approach for the lot sizing and schedulingproblem in the brewery industryrdquo Computers and IndustrialEngineering vol 72 no 1 pp 58ndash71 2014

[26] A Drexl and A Kimms ldquoLot sizing and schedulingmdashsurveyand extensionsrdquo European Journal of Operational Research vol99 no 2 pp 221ndash235 1997

[27] B Karimi SM T FatemiGhomi and JMWilson ldquoThe capac-itated lot sizing problem a review of models and algorithmsrdquoOmega vol 31 no 5 pp 365ndash378 2003

[28] R Jans and Z Degraeve ldquoMeta-heuristics for dynamic lot siz-ing a review and comparison of solution approachesrdquo EuropeanJournal of Operational Research vol 177 no 3 pp 1855ndash18752007

[29] R Jans and Z Degraeve ldquoModeling industrial lot sizing prob-lems a reviewrdquo International Journal of ProductionResearch vol46 no 6 pp 1619ndash1643 2008


[30] R Ramezanian M Saidi-Mehrabad and P Fattahi ldquoMIP for-mulation and heuristics for multi-stage capacitated lot-sizingand scheduling problem with availability constraintsrdquo Journalof Manufacturing Systems vol 32 no 2 pp 392ndash401 2013

[31] H Meyr ldquoSimultaneous lotsizing and scheduling on parallelmachinesrdquo European Journal of Operational Research vol 139no 2 pp 277ndash292 2002

[32] H Meyr ldquoSimultaneous lotsizing and scheduling by combininglocal search with dual reoptimizationrdquo European Journal ofOperational Research vol 120 no 2 pp 311ndash326 2000

[33] S Li X Zhong and H Li ldquoBatch delivery scheduling withmultiple decentralized manufacturersrdquoMathematical Problemsin Engineering vol 2014 Article ID 321513 7 pages 2014

[34] H Tempelmeier and L Buschkuhl ldquoDynamic multi-machinelotsizing and sequencing with simultaneous scheduling of acommon setup resourcerdquo International Journal of ProductionEconomics vol 113 no 1 pp 401ndash412 2008

[35] D Ferreira R Morabito and S Rangel ldquoSolution approachesfor the soft drink integrated production lot sizing and schedul-ing problemrdquo European Journal of Operational Research vol196 no 2 pp 697ndash706 2009

[36] D Ferreira A R Clark B Almada-Lobo and R MorabitoldquoSingle-stage formulations for synchronised two-stage lot sizingand scheduling in soft drink productionrdquo International Journalof Production Economics vol 136 no 2 pp 255ndash265 2012

[37] C F M Toledo L De Oliveira R De Freitas Pereira P MFranca and R Morabito ldquoA genetic algorithmmathematicalprogramming approach to solve a two-level soft drink produc-tion problemrdquo Computers amp Operations Research vol 48 pp40ndash52 2014

[38] C F M Toledo P M Franca R Morabito and A Kimms ldquoUmmodelo de otimizacao para o problema integrado de dimen-sionamento de lotes e programacao da producao em fabricas derefrigerantesrdquo Pesquisa Operacional vol 27 no 1 pp 155ndash1862007

[39] C F M Toledo P M Franca R Morabito and A KimmsldquoMulti-population genetic algorithm to solve the synchronizedand integrated two-level lot sizing and scheduling problemrdquoInternational Journal of Production Research vol 47 no 11 pp3097ndash3119 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Tanks

Productionlines

Products

Figure 1 The two-level production process

integer programming (MIP) models for lot sizing and sched-uling in the presence of setup times for problems found inelectrofused grain and animal nutrition industries Concern-ing multilevel lot sizing problems several publications exist(eg [7ndash10]) The present paper deals with a multilevel lotsizing and scheduling problem which is difficult to tackleeven if the capacity is unlimited [11 12] and there are parallelmachines represented by production lines (Figure 1) Thework in de Matta and Guignard [13] deals with rolling prod-uction schedules for lot sizingwith parallelmachines A studyon sequence-dependent setup costs with parallel machinesis found in Kang et al [14] A lot sizing problem in a semi-conductor assembly with parallel machines was described byKuhn and Quadt [15] and Quadt and Kuhn [16] A GRASPheuristic with path-relinking intensification was applied byNascimento et al [17] to solve instances of the capacitated lotsizing problem with parallel machines

A R Clark and S J Clark [18] proposed a MIP formu-lation for a lot sizing and scheduling problem on parallelmachines with sequence-dependent setup times Their com-plex MIP formulation was solved using approximate modelsin a rolling horizon basis with a reduced number of binaryvariables per period Model reformulations were also pre-sented by Wolsey [19] to make standard software applicableClark [20] presented a mathematical model for the planningof a canning line in a drinks manufacturer He proposedapproaches using local search integrated with the solution ofapproximate MIP models Gupta and Magnusson [21] pre-sented a model for the capacitated lot sizing and schedulingproblem with sequence-dependent setup costs and nonzerosetup timesThe solver CPLEXwas used to find optimal solu-tions for small sized instances The authors also determinedlower bounds for large sized instances using row aggregationand relaxing the integrality of somemodel variables Almada-Lobo et al [22] proposed models for the same problem tack-led by Gupta and Magnusson [21] which were solved usingspecific-purpose heuristics and the CPLEX solver

A synchronization problem arises in a two-level problemreported by Almada-Lobo et al [23] and Toledo et al [24]Their focus is a glass container industry where the first leveldeals with the glass color produced in the furnaces and

the second level deals with the color of the containers that areproduced afterwards MIP formulations are tailored to tacklethis two-level and synchronized production planning prob-lem where metaheuristic approaches are proposed Baldo etal [25] studied another two-level problem in the breweryindustry preparing the liquids including fermentation andmaturation inside the fermentation tanks and bottling theliquids on the filling linesThe problemdiffers from soft drinkproblemsmainly due to the relatively long lead times requiredfor the fermentation and maturation processes and becausethe ready liquid can remain in the tanks for some time beforebeing bottled The surveys in Drexl and Kimms [26] Karimiet al [27] Jans and Degraeve [28 29] and Ramezanian et al[30] can be referred to for complete overviews in lot sizingproblems and industrial modeling for lot sizing problems

Awork that comes close to our case is the one described inMeyr [31] which is an extension of Meyr [32] Meyr [31] con-sidered a lot sizing and scheduling problemwith nonidenticalparallel machines inventory costs production costs andsequence-dependent setup times and costs However the softdrink industry problem in the present paper is more complexin the sense that it has two lot sizing and scheduling problemseach of them similar to Meyrrsquos problem one for lines andanother for tanks The problem solution should find simulta-neously the lot sizing and scheduling of rawmaterials in tanksand soft drinks in the bottling lines because there is interde-pendence between these two levels which is not present inMeyrrsquos approach Thus new variables and constraints able tosynchronize and integrate these two levels need to be addedThere are studies dealing with supply chain synchronizationor integration applied to decentralized manufacturers [33]However the synchronization aspect dealt with by the presentpaper is closer to the work of Tempelmeier and Buschkuhl[34]These authors introduced a synchronization problem ina multi-item multimachine lot sizing and scheduling prob-lemThis problem is found in the production system of auto-mobile suppliers There is a predetermined assignment ofproducts to machines where a common resource is necessaryfor setup processes on all machines

Bearing in mind the mentioned aspects the problemstudied by the present paper is called synchronized and inte-grated two-level lot sizing and scheduling problem (SITLSP)Someworks relatedwith the specific softdrink industrial situ-ation considered by the SITLSP have been published recentlyFor instance in Ferreira et al [35] a simplified formulationwas presented for the SITLSP where each bottling line hasa dedicated tank and each tank can be filled with all liquidflavors needed by this line Single-stage formulations werepresented in Ferreira et al [36] based on the general lot sizingand scheduling problem (GLSP) and on the asymmetric trav-elling salesman problem (ATSP) The solutions found usingthese formulations outperformed those found by solving theformulation described in Ferreira et al [35] Toledo et al [37]introduced a genetic algorithmmathematical programmingapproach using Ferreira et alrsquos [35] model which reachedcompetitive or better solutions for the real-world instancessolved in Ferreira et al [36] All these recent results assumethe hypotheses of dedicated tanks In our formulation thereare no dedicated tanks as it will be explained later
















0 1 2 3 4 5 6 7 8 9


Time10

L1L2L3

P1 P2 P3P4P5P4

P6P5


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmA

RmD RmD

RmE

RmC

Tk1Tk2Tk3








RmA RmB

RmC

RmE

RmA

RmD RmC


RmD

Tk1

Tk2

Tk3

L1

L2

L3

P1

P4

P2 P3P1

P5 P5 P4

P6















0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmC

RmE

RmA

RmD RmD RmC

Tk1Tk2Tk3L1

L2

L3

P 1 P 2 P 3P 1

P 5 P 4

P 6


Micro-periods

P 5P 4






variables 119909119861119897119904120591 and 119909



119861119896119904120591 and 119909

119864119896119904120591 are used to











119869

sum

119894=1

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119879

sum

119905=1ℎ119895119868119895119905 +

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1V119895119897119902119895119897119904

+

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119871

sum

119896=1

119879

sum

119905=1ℎ119895119868119895119896119905sdot119879119898

+

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1V119895119897119902119895119896119904 + 119872

119869

sum

119895=11199020119895

(1)


119909119895119897119904 = 0

119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(2)

119869

sum

119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)

119901119895119897119902119895119897119904 le 119862119909119895119897119904

119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(4)

1198681198951 = 1198681198950 + 1199020119895 +

119871

sum

119897=1

119878

sum

119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)

119868119895119905 = 119868119895119905minus1 +

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905

119895 = 1 119869 119905 = 2 119879

(6)

119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(7)

120590119897119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119897119911119894119895119897119904

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(8)

120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)

119905sdot119878

sum

119904=(119905minus1)119878+1(120590119897119904 +

119869

sum

119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905

119897 = 1 119871 119905 = 1 119879

(10)


0119895 Constraint



119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119905sdot119878120591 ge 120596119897119905+1

119897 = 1 119871 119905 = 1 119879 minus 1

(11)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904minus1120591 ge 120590119897119904

+

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(12)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898


+

119869

sum

119895=1119901119895119897119902119895119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

(13)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(14)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(15)

119869

sum

119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)

120576119906119897119904 le

119869

sum

119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(18)



0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)



s4

pP2L1qP2L1s3


119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum


119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)




119861119897119904120591 = 119909






119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























0 1 2 3 4 5 6 7 8 9


Time10

L1L2L3

P1 P2 P3P4P5P4

P6P5


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmA

RmD RmD

RmE

RmC

Tk1Tk2Tk3








RmA RmB

RmC

RmE

RmA

RmD RmC


RmD

Tk1

Tk2

Tk3

L1

L2

L3

P1

P4

P2 P3P1

P5 P5 P4

P6















0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmC

RmE

RmA

RmD RmD RmC

Tk1Tk2Tk3L1

L2

L3

P 1 P 2 P 3P 1

P 5 P 4

P 6


Micro-periods

P 5P 4






variables 119909119861119897119904120591 and 119909



119861119896119904120591 and 119909

119864119896119904120591 are used to











119869

sum

119894=1

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119879

sum

119905=1ℎ119895119868119895119905 +

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1V119895119897119902119895119897119904

+

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119871

sum

119896=1

119879

sum

119905=1ℎ119895119868119895119896119905sdot119879119898

+

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1V119895119897119902119895119896119904 + 119872

119869

sum

119895=11199020119895

(1)


119909119895119897119904 = 0

119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(2)

119869

sum

119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)

119901119895119897119902119895119897119904 le 119862119909119895119897119904

119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(4)

1198681198951 = 1198681198950 + 1199020119895 +

119871

sum

119897=1

119878

sum

119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)

119868119895119905 = 119868119895119905minus1 +

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905

119895 = 1 119869 119905 = 2 119879

(6)

119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(7)

120590119897119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119897119911119894119895119897119904

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(8)

120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)

119905sdot119878

sum

119904=(119905minus1)119878+1(120590119897119904 +

119869

sum

119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905

119897 = 1 119871 119905 = 1 119879

(10)


0119895 Constraint



119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119905sdot119878120591 ge 120596119897119905+1

119897 = 1 119871 119905 = 1 119879 minus 1

(11)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904minus1120591 ge 120590119897119904

+

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(12)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898


+

119869

sum

119895=1119901119895119897119902119895119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

(13)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(14)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(15)

119869

sum

119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)

120576119906119897119904 le

119869

sum

119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(18)



0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)



s4

pP2L1qP2L1s3


119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum


119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)




119861119897119904120591 = 119909






119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmC

RmE

RmA

RmD RmD RmC

Tk1Tk2Tk3L1

L2

L3

P 1 P 2 P 3P 1

P 5 P 4

P 6


Micro-periods

P 5P 4






variables 119909119861119897119904120591 and 119909



119861119896119904120591 and 119909

119864119896119904120591 are used to











119869

sum

119894=1

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119879

sum

119905=1ℎ119895119868119895119905 +

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1V119895119897119902119895119897119904

+

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119871

sum

119896=1

119879

sum

119905=1ℎ119895119868119895119896119905sdot119879119898

+

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1V119895119897119902119895119896119904 + 119872

119869

sum

119895=11199020119895

(1)


119909119895119897119904 = 0

119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(2)

119869

sum

119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)

119901119895119897119902119895119897119904 le 119862119909119895119897119904

119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(4)

1198681198951 = 1198681198950 + 1199020119895 +

119871

sum

119897=1

119878

sum

119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)

119868119895119905 = 119868119895119905minus1 +

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905

119895 = 1 119869 119905 = 2 119879

(6)

119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(7)

120590119897119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119897119911119894119895119897119904

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(8)

120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)

119905sdot119878

sum

119904=(119905minus1)119878+1(120590119897119904 +

119869

sum

119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905

119897 = 1 119871 119905 = 1 119879

(10)


0119895 Constraint



119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119905sdot119878120591 ge 120596119897119905+1

119897 = 1 119871 119905 = 1 119879 minus 1

(11)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904minus1120591 ge 120590119897119904

+

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(12)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898


+

119869

sum

119895=1119901119895119897119902119895119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

(13)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(14)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(15)

119869

sum

119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)

120576119906119897119904 le

119869

sum

119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(18)



0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)



s4

pP2L1qP2L1s3


119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum


119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)




119861119897119904120591 = 119909






119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869

sum

119894=1

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119879

sum

119905=1ℎ119895119868119895119905 +

119869

sum

119895=1

119871

sum

119897=1

119879sdot119878

sum

119904=1V119895119897119902119895119897119904

+

119869

sum

119894=1

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1119904119894119895119897119911119894119895119897119904 +

119869

sum

119895=1

119871

sum

119896=1

119879

sum

119905=1ℎ119895119868119895119896119905sdot119879119898

+

119869

sum

119895=1

119871

sum

119896=1

119879sdot119878

sum

119904=1V119895119897119902119895119896119904 + 119872

119869

sum

119895=11199020119895

(1)


119909119895119897119904 = 0

119895 = 1 119869 119897 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(2)

119869

sum

119895=1119909119895119897119904 = 1 119897 = 1 119871 119904 = 1 119879 sdot 119878 (3)

119901119895119897119902119895119897119904 le 119862119909119895119897119904

119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(4)

1198681198951 = 1198681198950 + 1199020119895 +

119871

sum

119897=1

119878

sum

119904=1119902119895119897119904 minus 1198891198951 119895 = 1 119869 (5)

119868119895119905 = 119868119895119905minus1 +

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119897119904 minus 119889119895119905

119895 = 1 119869 119905 = 2 119879

(6)

119911119894119895119897119904 ge 119909119895119897119904 + 119909119894119897119904minus1 minus 1

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(7)

120590119897119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119897119911119894119895119897119904

119894 119895 = 1 119869 119897 = 1 119871 119904 = 1 119879 sdot 119878

(8)

120596119897119905 le 120590119897(119905minus1)119878+1 119897 = 1 119871 119905 = 1 119879 (9)

119905sdot119878

sum

119904=(119905minus1)119878+1(120590119897119904 +

119869

sum

119895=1119901119895119897119902119895119897119904) le 119862 + 120596119897119905

119897 = 1 119871 119905 = 1 119879

(10)


0119895 Constraint



119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119905sdot119878120591 ge 120596119897119905+1

119897 = 1 119871 119905 = 1 119879 minus 1

(11)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904minus1120591 ge 120590119897119904

+

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(12)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898


+

119869

sum

119895=1119901119895119897119902119895119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

(13)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(14)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119897119904120591 = 1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(15)

119869

sum

119895=1119901119895119897119902119895119897119904 le 119862119906119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (16)

120576119906119897119904 le

119869

sum

119895=1119902119895119897119904 119897 = 1 119871 119904 = 1 119879 sdot 119878 (17)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(18)



0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)



s4

pP2L1qP2L1s3


119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum


119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)




119861119897119904120591 = 119909






119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time

Time

L1

L1s1 s2 s3 s4

P1 P1 P2 P3

P1 P1 P2 P3

xEL1s38

xEL1s38

xEL1s41012059012

120590L1s3120596L12

Constraint (12)

Constraint (13)



s4

pP2L1qP2L1s3


119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 le 119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(19)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904minus1120591 le 119879

119898119906119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 2 119879 sdot 119878

(20)

119905sdot119879119898

sum


119898+ 1) le 119879

119898119906119897(119905minus1)119878+1

119897 = 1 119871 119905 = 1 119879

(21)

119906119897119904 ge 119906119897119904+1

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878 minus 1

(22)

120576119906119897119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591

minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 le

119869

sum

119895=1119901119895119897119902119895119897119904

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(23)

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119897119904120591 ge

119869

sum

119895=1119901119895119897119902119895119897119904

minus 119862119898

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1) 119878 + 1 119879 sdot 119878

(24)




119861119897119904120591 = 119909






119902119895119897119904 =

119871

sum

119896=1

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119896119895119897119904120591 (25)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

120575119897119904 = (

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119897119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119897119904minus1120591 + 1) 119862

119898

minus

119869

sum

119895=1119901119895119897119902119895119897119904

(26)


xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










xBL1s37 xEL1s38

xBL1s49 xEL1s49

120590L1s2s3 s4




9xEL1s49 minus 9

minus 1


Macro-period 2

Constraint (23)

Constraint (24)

L1 P2 P3

6 7 8 9 10

Time



Constraint (29)

Tk1

L1

L2

s3

s3

s3

RmA

P2

P3

xBL2s37

xEL1s37

xBL1s37

xEL2s38







Time


119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le 119862

119898minus 120575119897119904 + (1minus 119909

119861119897119904120591) 119862119898

(27)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198641198971199041205911015840 (28)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119871

sum

119896=1

119869

sum

119895=1119901119895119897119902119896119895119897119904120591 le

119905sdot119879119898

sum

1205911015840=(119905minus1)119879119898+1119862119898

1199091198611198971199041205911015840 (29)

119897 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119879 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898






119909119895119896119904 = 0

119895 = 1 119869 119896 isin 1 119871 119871119895 119904 = 1 119879 sdot 119878

(30)

119869

sum

119895=1119909119895119896119904 = 1 119896 = 1 119871 119904 = 1 119879 sdot 119878 (31)

119902119895119896119904 le 119876119896119909119895119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(32)

119902119895119896119904 le 119876119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(33)

119869

sum

119895=1119902119895119896119904 ge 119876

119896119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(34)

119911119894119895119896119904 ge 119909119895119896119904 + 119909119894119896119904minus1 minus 2+ 119906119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(35)

119909119895119896119904 minus 119909119895119896119904minus1 le 119906119896119904

119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(36)


Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Tk1

Tk2

RmC

RmB RmB

RmDs3 s4

6 7 8 9 10

Time


Constraint (35)

Constraint (36)

Macro-period 2









120590119896119904 =

119869

sum

119894=1

119869

sum

119895=11205901015840119894119895119896119911119894119895119896119904

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(37)

120596119896119905119904 le 120590119896(119905minus1)119878+1

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(38)

119905sdot119878

sum

119904=(119905minus1)119878+1120590119896119904 le 119862 + 120603119896119905

119894 119895 = 1 119869 119896 = 1 119871 119904 = 1 119879 sdot 119878

(39)





119902119895119896119904 =

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119902119895119896119904120591 (40)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 119904 = 1 119879 sdot 119878

119905 sdot 119862 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896119905sdot119878120591ge 120603119896119905+1 (41)

119896 = 1 119871 119905 = 1 119879 minus 1119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904minus1120591 ge 120590119896119904 (42)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591


(43)

119896 = 1 119871 119905 = 1 119879119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119864119896119904120591 = 1 (44)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119909119861119896119904120591 = 1 (45)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119909119861

119896(119905minus1)119878+1120591 = 1 (46)

119896 = 1 119871 119905 = 2 119879


119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119879119898

sum

120591=11199091198611198961120591 = 1 (47)

119896 = 1 119871 119905 = 2 119879

119869

sum

119895=1119902119895119896119904120591 le 119876119896119909

119864119896119904120591 (48)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 120591 =

(119905 minus 1)119879119898

+ 1 119905 sdot 119879119898

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 le 119879

119898119906119896119904 (49)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591

le 119879119898

119906119896(119905minus1)119878+1

(50)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198641198961120591 minus119879119898

sum

120591=11205911199091198611198961120591 le 119879

1198981199061198961 (51)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904120591 le 119906119896119904 (52)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1120591119909119861

119896(119905minus1)119878+1120591 minus119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864

119896(119905minus1)119878+1120591

le 119906119896(119905minus1)119878+1

(53)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus119879119898

sum

120591=11205911199091198641198961120591 le 1199061198961 (54)

119896 = 1 119871

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119864119896119904minus1120591 le 119879

119898119906119896119904 (55)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1120591119909119861


+ 1)

le 119879119898

119906119896(119905minus1)119878+1

(56)

119896 = 1 119871 119905 = 2 119879

119879119898

sum

120591=11205911199091198611198961120591 minus 1 le 119879

1198981199061198961 (57)

119896 = 1 119871

119906119896119904 ge 119906119896119904+1 (58)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 1 119905 sdot 119878 minus 1

119862119898

119906119896119904 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864119896119904120591 minus

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119861119896119904120591

= 120590119896119904

(59)

119896 = 1 119871 119905 = 1 119879 119904 = (119905 minus 1)119878 + 2 119905 sdot 119878

119862119898

119906119896(119905minus1)119878+1 +

119905sdot119879119898

sum

120591=(119905minus1)119879119898+1119862119898

120591119909119864

119896(119905minus1)119878+1120591

minus

119905sdot119879119898

sum

120591=(119905minus2)119879119898+1119862119898

120591119909119861

119896(119905minus1)119878+1120591 = 120590119896(119905minus1)119878+1

(60)

119896 = 1 119871 119905 = 2 119879

119862119898

1199061198961 +

119879119898

sum

120591=1119862119898

1205911199091198641198961120591 minus119879119898

sum

120591=1119862119898

1205911199091198611198961120591 = 1205901198961 minus 1205961198961 (61)

119896 = 1 119871

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (62)


119898+ 1 119905 sdot 119879

119898

119869

sum

119895=1119868119895119896120591minus1 le 119876119896 ((1minus 119909

119861119896119904120591) + (1minus 119906119896119904)) (63)

119896 = 1 119871 119904 = (119879 minus 1)119878 + 1 119879 sdot 119878 + 1 120591 = (119879 minus 1)119879119898

+

1 119879 sdot 119879119898


119864Tk211990437 = 119909




Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Tk 2

6 7 8 9 10

Time

Macro-period 2

RmC RmB

xETk2s37xETk2s49












119868119895119896120591 = 119868119895119896120591minus1 +

119905sdot119878

sum

119904=(119905minus1)119878+1119902119895119896119904120591

minus

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591

(64)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898

119868119895119896120591 ge

119869

sum

119894=1

119871

sum

119897=1

119905sdot119878

sum

119904=(119905minus1)119878+1119903119895119894119902119896119894119897119904120591+1 (65)

119895 = 1 119869 119896 = 1 119871 119905 = 1 119879 120591 = (119905minus1)119879119898

+1 119905 sdot

119879119898



0 101 2 3 4 5 6 7 8 9Time

RmA RmBRmC

RmE

RmARmD RmD RmC


Tk1Tk2Tk3L1

L3

P1 P2 P3P1

P5P5 P4

P6

L2

Micro-periods

P4


0 101 2 3 4 5 6 7 8 9Time

RmA RmB

RmCRmE

RmA

RmD RmD RmCs4

s4

s4

s3Tk1

Tk2

Tk3

L1

L3

P1

P4

P2 P3P1

P5P5 P4

P6

L2









119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898 5 119876119896

1000 l 119876119896 5000 l119862 5 tu 119862

119898 1 tu min119863119890119898 500 u max119863119890119898 10000 u


Comb 119871119871119869119869 Comb 119871119871119869119869 Comb 119871119871119869119869

1198781 2221 1198784 3321 1198787 43211198782 2232 1198785 3332 1198788 43321198783 2242 1198786 3342 1198789 4342



119861119897119904120591 119909119864119897119904120591 119909119861119896119904120591 and 119909

119864119896119904120591









Gap() = 100(119885 minus 119885

119885

) (66)






1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1205901015840



119904119894119895119896 119904119894119895119896 = 119891 sdot 1205901015840




Comb 119879 = 1 119879 = 2



Comb119879 = 1 119879 = 2


1198781 100 263 281 210 533 5751198782 136 499 454 282 1004 9191198783 142 615 512 294 1235 10391198784 150 452 419 315 916 8621198785 204 880 673 423 1771 13681198786 213 1098 764 441 2206 15521198787 176 555 498 367 1122 10131198788 246 1100 821 507 2211 16411198789 258 1390 924 531 2790 1865








Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Comb 119879 = 3 119879 = 4



Comb 119879 = 3 119879 = 4



Comb 119871119871119869119869119879 Comb 119871119871119869119869119879

1198711 551054 1198716 5615881198712 561584 1198717 8510581198713 851054 1198718 8615881198714 861584 1198719 55105121198715 551058 11987110 5615812






Comb 1198771 1198772





0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

500

1000

1500

2000

2500

3000

Aver

age C

PU (s

)








6 Conclusions





Appendix


General Parameters



(119862 = 119879119898

times 119862119898)











product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













product 119895 (1205901015840119894119895119897 le 119862 and 1205901015840119894119895119897 = 0)




















Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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