research article assembly line productivity assessment by

Research ArticleAssembly Line Productivity Assessment by ComparingOptimization-Simulation Algorithms of Trajectory Planning forIndustrial Robots

Francisco Rubio Carlos Llopis-Albert Francisco Valero and Josep Lluiacutes Suntildeer

Centro de Investigacion en Ingenierıa Mecanica (CIIM) Universitat Politecnica de Valencia-Camino de Vera sn46022 Valencia Spain

Correspondence should be addressed to Carlos Llopis-Albert cllopisagmailcom

Received 25 July 2014 Revised 20 September 2014 Accepted 22 September 2014

Academic Editor Shaofan Li

Copyright copy 2015 Francisco Rubio et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper an analysis of productivity will be carried out from the resolution of the problem of trajectory planning ofindustrial robots The analysis entails economic considerations thus overcoming some limitations of the existing literature Twomethodologies based on optimization-simulation procedures are compared to calculate the time needed to perform an industrialrobot task The simulation methodology relies on the use of robotics and automation software called GRASP The optimizationmethodology developed in this work is based on the kinematics and the dynamics of industrial robots It allows us to pose amultiobjective optimization problem to assess the trade-offs between the economic variables by means of the Pareto fronts Thecomparison is carried out for different examples and from a multidisciplinary point of view thus to determine the impact ofusing each method Results have shown the opportunity costs of non using the methodology with optimized time trajectoriesFurthermore it allows companies to stay competitive because of the quick adaptation to rapidly changing markets

1 Introduction

Time needed to perform a trajectory for industrial robots isa very important issue in order to improve productivity inmany economic activities Specifically most algorithms seekto find the minimum time trajectory in order to increase theworking time and subsequently to reduce the unproductivetime The existing literature shows a lack of studies that con-sider both the economic issues and the motion of industrialrobots

In this paper the working times of industrial robots arecompared between twodifferent approacheswhile taking intoaccount the corresponding economic impacts The compari-son is applied to several examples which covers a wide rangeof parameters that govern the kinematics and dynamics of theindustrial robotsThe first methodology is based on a roboticsimulation program called GRASP (BYG System Ltd) and thesecond on optimization techniques

When the working times have been calculated theassembly line productivity is estimated by means of the time

difference so that we can quantify the impact of eachmethodProductivity is quantified by conducting an economic studybased on the working times of robotic tasks and morespecifically the time needed to manufacture and assemble acertain product

A multiobjective optimization problem is posed to assessthe trade-offs between the economic variables by means ofthe Pareto fronts (see Section 6) These fronts will serve todetermine those variables that mostly influence the increaseof productivity of the assembly line We will prove thatworking times (not working cycles) are critical from aneconomic point of view and so are the methods to obtainthem

Those times will enable us to set conclusions about whichmethod is more useful in order to increase the productivityof the robotic system

The paper is organized as follows Initially we will explainin detail the main characteristics of the trajectory planningmethodology and how the time is obtained Consequentlythe economic analysis will provide insight on the productivity

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 931048 10 pageshttpdxdoiorg1011552015931048

2 Mathematical Problems in Engineering

of assembly lines Finally the conclusions will be discussed inthe last section

2 Background of the TrajectoryPlanning Problem

Currently there are a great number of methodologies to solvethe trajectory planning problem for industrial robots whichgive the time needed to perform a task But few papers tacklethe analysis of productivity related to the working timesobtained

Over the years the algorithms have been polished andthe working assumptions of the robotic systems have beenincreasingly adjusted to real conditions This fact has beenachieved by analysing the complete behaviour of the roboticsystem particularly the characteristics of the actuators andthe mechanical structure of the robot To tackle this problemother important working parameters and variables have beentaken into account such as the input torques the energyconsumed and the power transmitted Furthermore thekinematic properties of the robotrsquos links such as the veloc-ities accelerations and jerks must be also considered Theaforementioned algorithms provide a smooth robot motionfor the robotic system

To obtain the best trajectories in terms of minimumtimes some of the working parameters have been includedin the appropriate objective function of the optimizationprocedure (input torques the energy consumed and thepower transmitted) The optimization criteria most widelyused can be sorted as follows

(1) Minimum time required which is directly boundedto productivity

(2) Minimum jerk which is bounded to the quality ofwork accuracy and equipment maintenance

(3) Minimum energy consumed or minimum actuatoreffort both linked to savings

(4) Hybrid criteria for example minimum time andenergy

In the past the early algorithms that solved the trajectoryplanning problem tried to minimize the time needed forperforming the task (see [1ndash3]) One disadvantage of thoseminimum-time algorithms was that the trajectories haddiscontinuous values of acceleration and torques which led todynamic problems during the trajectory performance Thoseproblems were avoided by imposing smooth trajectories tobe followed such as spline functions which have been usedin both path and trajectory planning

The early algorithms in trajectory planning sought tominimize the time needed for performing the task Thedynamics properties of actuators were neglected A recentexample of this type of algorithm can be found in [4] whichdetermines smooth and near time-optimal path-constrainedtrajectories It considers not only velocity and acceleration butalso jerk

Later the trajectory planning problem was tackled bysearching for jerk-optimal trajectories Jerks are highly

important for working with precision and without vibrationThey also have an effect on the control system and thewearingof mobile parts such as joints and bars These methods allowa reduction in errors during trajectory tracking the stressesin the actuators and also in the mechanical structure ofthe robot and the excitement of resonance frequencies Jerkrestriction is introduced by other authors [5 6]

In [7] a method is introduced for determining smoothand time-optimal path-constrained trajectories for roboticmanipulators by imposing limits on the actuator jerks

In [8] a global minimum-jerk trajectory planning algo-rithm of a space manipulator is presented

Another different approach to solving the trajectoryplanning problem is based on minimizing the torque andthe energy consumed instead of the trajectory time or thejerk This approach leads to smoother trajectories An earlyexample is seen in [9]

Similarly in [10] the authors searched for the minimumenergy consumed They proposed a method for solvingthe trajectory generation problem in redundant degree offreedommanipulatorsThey used a variational approach andthe B-Spline curve was introduced to minimize the electricalenergy consumed in a robot manipulator system

Thework in [11] also takes into account energyminimiza-tion for the trajectory planning problem

In [12] the authors proposed a technique of iterativedynamic programming to plan minimum energy consump-tion trajectories for robotic manipulators The dynamicprogramming method was modified to perform a series ofdynamic programming passes over a small reconfigurablegrid covering only a portion of the solution space at anyone pass Although strictly no longer a global optimizationprocess this iterative approach retained the ability to avoidcertain poor local minima while avoiding the dimensionalissue associatedwith a pure dynamic programming approachThe modified dynamic programming approach was veri-fied experimentally by planning and executing a minimumenergy consumed path for a Reis V15 industrial manipulator

Afterwards new perspectives appear for solving thetrajectory planning problem The main point was to use aweighted objective function to optimize the working param-eters [13] There the cost function is a weighted balance oftransfer time the mean average of the torques and power

In this paper we will introduce two methods to solve thetrajectory planning problem for industrial robots workingin complex environments The time will be used in theeconomical study

In the first method the procedure calculates the optimaltrajectory by neglecting initially the potential presence ofobstacles in the workspace By removing the obstacles (realor potential) from the optimization problem the algorithmwill calculate a minimum time trajectory as a starting pointThen the procedure must take into account the real obstaclespresented in the workspace When obstacles are consideredthe initial trajectory will not be feasible and will have toevolve so that it can become a solution The way this initialtrajectory evolves until a new feasible collision-free trajectoryis obtained is presented in this paper It is a direct algorithmthat works in a discrete space of trajectories approaching the

Mathematical Problems in Engineering 3

first solution to the global solution as the discretization isrefined The solutions obtained are efficient trajectories (iethe minimum time trajectories) All the trajectories obtainedmeet the physical limitations of the robot The solution alsoavoids collisions and takes into account the constraint ofenergy consumed

The second method calculates the times using the kine-matic properties of the robotic system by means of a simula-tion program called GRASP

3 Time Obtained Using the ProposedOptimization Trajectory Planner

Our objective is to calculate the minimum time trajectory (119905)between the initial and final configurations Any robot con-figuration119862119895 = 119862119895(120572119895119894 119901

119895

119896) can be expressed unequivocally by

means of the Cartesian coordinates of significant points of therobot 120572119895119894 = (120572

119895

119909119894 120572119895

119910119894 120572119895

119911119894)We calculate the time needed to go from 119862119894 to 119862119891

This process is based on an optimization problem to obtainthe minimum time between these two configurations Theproblem is transformed into obtaining the minimum timeover an interpolated trajectory between both configurationssubjected to physical constraints in the actuators

This optimization problem can be stated as in [7] asfollows

Find 119902 (119905) 120591 (119905) 119905119891 (1)

between each of the two configurations (see Section 33)

Minimizing min120591isinΩ119869 = int

119905119891

0

119889119905 (2)

where 120591(119905) isin 119877119899 is the vector of the actuator torques andΩ isthe space state in which the vector of the actuator torques isfeasible

The optimization problem is subject to

(1) the robot dynamics

119872(119902 (119905)) 119902 (119905) + 119862 (119902 (119905) 119902 (119905)) 119902 (119905) + 119892 (119902 (119905)) = 120591 (119905) (3)

(2) unknown boundary conditions (position velocityand acceleration) for intermediate configurations apriori

119902 (119905intminus1) = 119902intminus1 119902 (119905int) = 119902int

119902 (119905intminus1) = 119902intminus1 119902 (119905int) = 119902intminus1


(4)

(3) boundary conditions for initial and final configura-tions

119902 (0) = 119902119900 119902 (119905119891) = 119902119891

119902 (0) = 0 119902 (119905119891) = 0

(5)

(4) collision avoidance within the robot workspace

119889119894119895 ge 119903119895 + 119908119894 (6)

119889119894119895 being the distance from any obstacle 119895 (spherecylinder or prism) to robot arm 119894 119903119895 is the character-istic radius of the obstacle and 119908119894 is the radius of thesmallest cylinder that contains the arm 119894

(5) actuator torque rate limits

120591min119894 le 120591119894 (119905) le 120591

max119894 forall119905 isin [0 119905min] 119894 = 1 dof (7)

(6) maximum power in the actuators

119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875

max119894 forall119905 isin [0 119905min] 119894 = 1 dof

(8)

(7) maximum jerk on the actuators

119902min119894 le

119902119894 (119905) le

119902max119894 forall119905 isin [0 119905min] 119894 = 1 dof

(119902119894 is the jerk of actuator 119894)

(9)

(8) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (10)

where 120576119894119895 is the energy consumed by the actuator 119894 betweenthe configurations 119888119895 and 119888119895+1

Here the main definitions and processes used to obtainthe free-collision trajectories of the robot (and subsequentlythe time needed to perform the trajectory) are detailed

31 Robot Configuration It is expressed in joint coordinates119888119896(119902119894) with a view to define kinematics and dynamics of therobot When dealing with collisions Cartesian coordinates119888119896(120582119895) will be used being 119894 = 1 dof 119895 = 1 npc dofrobot degrees of freedom npc number of Cartesian pointsused for the wired model of the robot in collision detectionand 119896 is the configuration itself see [14ndash16]

32 Adjacent Configuration Given a feasible configuration ofthe robot 119888119896 it is said that 119888119897 is adjacent to it if it is feasible andmeets the following two conditions

(i) The robot end-effector occupies a position corre-sponding to a node of the discretized workspace inCartesian coordinates and its distance to the end-effector position in the configuration 119888119896 is less thana given value

(ii) 119888119897 is such that it minimizes the function

dofsum119894=1

(119902119897119894 minus 119902119896119894 )2 (11)


33 Trajectory Given a sequence of 119898 robot configurations= 1198881(1199021119894 ) 119888

2(1199022119894 ) 119888119898(119902119898119894 ) the trajectory 119904 is defined by

means of cubic interpolation functions between adjacentconfigurations so that the resulting time 119905min to perform thetrajectory is minimum We have that

forall119905 isin [119905119895minus1 119905119895[ 997888rarr 119902119894119895 = 119886119894119895 + 119887119894119895119905 + 1198891198941198951199052+ 1198901198941198951199053

(12)

where 119894 = 1 dof 119895 = 1 119898 minus 1To ensure continuity the following conditions associated

with the given configurations are considered

(a) Position for each interval 119895 the initial and finalpositions must match 119888119895 and 119888119895+1 this gives a total of(2 dof (119898 minus 1)) equations

119902119894119895 (119905119895minus1) = 119902119895

119894

119902119894119895 (119905119895) = 119902119895+1

119894 (13)

(b) Velocity the initial and final velocities of thetrajectory must be zero obtaining (2 dof) equations

1199021198941 (1199050) = 0

119902119894119898minus1 (119905119898minus1) = 0(14)

When passing through each intermediate configura-tion the final velocity of previous interval must beequal to the initial velocity of the next interval thatgives (dof (119898 minus 2)) equations

119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (15)

(c) Acceleration for each intermediate configurationthe final actuator acceleration of the previous intervalmust be equal to the initial acceleration of the nextresulting in (dof (119898 minus 2)) equations

119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (16)

Knowing the time required to perform the trajectorybetween the different configurations using the aboveequations the coefficients of the cubic polynomialscan be obtained efficiently bymeans of the calculationof the normal time [15]In addition theminimum time trajectory smustmeetthe following four types of constraints(d) maximum torque in the actuators(e)

120591min119894 le 120591119894 (119905) le 120591


(f) maximum power in the actuators(g)

119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(18)

(h) maximum jerk on the actuators(i)

119902min119894 le

119902119894 (119905) le

119902max119894 forall119905 isin [0 119905min] 119894 = 1 dof (19)

(j) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (20)

where 120576119894119895 is the energy consumed by the actuator 119894between the configurations 119888119895 and 119888119895+1

To obtain the minimum time an optimization problemis solved using variables defined in time increment at eachinterval (see [17]) so that in the interval between 119888119895 and 119888119895+1the variable is Δ119905119895 = 119905119895 minus 119905119895minus1 and the objective function is

119898minus1

sum119895=1

Δ119905119895 = 119905min (21)

34 Offspring Trajectory Let 119904119895 be aminimum time trajectoryassociated to the sequence of 119898 configurations 119862119895 under theconditions described in Section 35 It is said that the trajec-tory 119904119896 is an offspring of 119904119895 when the following conditions aremet

(a) 119862119896 = 119862119895 cup 119888119899(b) 119899 = 1(c) 119899 = 119898 + 1

So the trajectories of a certain generation will have onepassing configurationmore than the previous generation butthey will keep the same initial and final configurations

35 Obtaining of the Collision-Free Trajectory The problemof obtaining a feasible and efficient trajectory for a robot inan environment with static obstacles allowing the motionbetween two given configurations (119888119894 and 119888119891) is posed Anefficient trajectory is that performed in a minimum timewith a reasonable computational cost and subject to thelimitations of the robot dynamics the jerk constraints andpower consumption Clearly the feasibility of the trajectorymeans that there are no collisions

Theproposed process for solving the problem involves thefollowing steps which are implemented in the algorithm

(a) Obtaining the minimum time trajectory using theprocedure described in Section 33 the trajectory119904min is obtained corresponding to the sequence ofconfigurations 119862 = 119888119894 119888119891

(b) Search for collisions the first configuration from 119904minwhich has collision 119888119888 is determined and a previousconfiguration 119888119886 is searched for whose distance isless than 119889seg (so that the smallest patterned obstacleused to represent the work environment can never bebetween 119888119888 y 119888119886)


Table 1 Kinematic characteristics of PUMA 560 robot

Joint 1 2 3 4 5 6Minimum angle (∘) minus1600 minus2150 minus450 minus1400 minus1000 minus2660Maximum angle (∘) 1600 350 2250 1400 1000 2660Maximum velocity (∘s) 820 540 1220 2280 2410 2280

(c) Obtaining adjacent configurations up to six newadjacent configurations to 119888119886 can be achieved asdefined in Section 32 (119888119886119895 119895 = 1 6)

(d) Obtaining offspring trajectories for each one of the119897 adjacent configurations obtained in the previoussection that have no collision with obstacles theoffspring trajectory 119904119896 is obtained from 119904min such that119862119896 = 119862 cup 119888119886119896 (119896 = 1 119897)

(e) Trajectory selection the generated trajectories areintroduced in previous section (d) on the set oftrajectories ordered by time 119879119905 = 1199041 sdot sdot sdot 119904119901 takingthe minimum time trajectory 1199041 and checking for nocollisions as it was done in previous section (b) If1199041 has no collision the algorithm goes to the nextsection otherwise it returns to section (c) and theprocess is repeated

(f) Refining the trajectory in case that the collision-free trajectory 1199041 does not belong to the first gen-eration (direct offspring of 119904min with a sequence ofthree configurations) we have 1199041 such that 1198621 =119888119894 1198882 1198883 119888119898minus1 119888119891 (119898 being the number of con-figurations that define the trajectory)

119898 minus 2 sets of configurations 1198621119901 are taken such that 1198621 =1198621119901cup119888119901 for119901 = 2 119898minus1 obtaining the corresponding set of

collision-free trajectories119879119903 If it is empty then it is said that 1199041cannot be reduced otherwise the process is repeated for thenew trajectories and the results are included in119879119903Theprocessfinishes when the algorithm cannot obtain new trajectories

Finally the trajectory 1199041 is included in 119879119903 and the reducedtrajectory 119904119903 is defined as the trajectory belonging to 119879119903 withminimum time

The proposed solution to the problem is 119904119903 which will bea minimum time offspring trajectory 119904min and with a smallnumber of passing configurations

4 Time Obtained Using a RoboticEnvironment Simulation ProgramCalled GRASP

The simulation program GRASP10 for robotic environ-ments is used to obtain the time needed to perform themotion between two given configurations Among othertasks GRASP10 can model and simulate the robot kinematicbehavior In this paper we have used the original modelof PUMA 560 robot that comes with the program as acomparator It is assumed that the point to point trajec-tory calculation procedures of GRASP10 correspond to the

real robot The robot kinematic characteristics are shownin Table 1

It should be noted that GRASP10 does not performdynamic calculations but only kinematic ones It is thereforevery important to indicate the maximum working velocitieswhich have been obtained from actual robot by consideringthe properties of each actuator primarily its maximumpower and the working torque

When the trajectory is generated by GRASP10 the work-ing actuators act simultaneously and at least one of them ismoved to its maximum speed calculated as follows for eachactuator (see Table 1)

120596max =119875

119905min (22)

where 119875 stands for the power and 119905min for the torque in thecorresponding actuator

This methodology (use of GRASP10 for calculating thetime required to perform a trajectory) has been applied to thesame examples that have been resolved by the optimizationalgorithm explained in Section 3 in order to compare theirefficiency

For each example the initial data are the initial and finalconfigurations and the kinematics of the robot The obstaclehas been incorporated after the generation of the path so thatit burdens the previously calculated one

5 Productivity and Economic Study

In this section the productivity will be quantified by conduct-ing an economic study based on the working times of robotictasks

The aim is to increase the profitability of productionlines by designing flexiblemanufacturing systemsThis allowscompanies to stay competitive because of the quick adapta-tion to rapidly changing markets For instance by adjustingthe working hours in assembly lines or by deciding whichproducts are more suitable to be manufactured according tothe current demand

This is performed by posing a multiobjective optimiza-tion problem which makes use of the optimization algo-rithm above presented to solve the kinematics and dynamicsof robot arms The optimization method finds the minimumtime trajectory to perform industrial tasks in production lineswhile taking into consideration the physical constraints ofthe real posed problem and then economic issues are alsoconsidered in the process

Furthermore Pareto fronts will be introduced which willserve to determine those variables which mainly affect theimproved productivity To bemore precise themultiobjectiveoptimization problem allows obtaining the Pareto frontiers


which provides information about the trade-offs betweenthe competing variables (ie execution times and benefitsfor the different products that can be manufactured at theproduction line)

Therefore the economic study starts by defining theeconomic objective function to be used It is formulated asfollows

Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)

where 119861 is the objective function to be maximized andrepresents the current value of the net benefit from ageneric product (in C) defined as the revenue of the itemsmanufactured at a production line minus total costs 119903 is theannual discount rate 119879 represents number of years 119875119901 is themarket unitary price of the product 119901 (in C) 119862119901 stands forthe unitary cost to perform the product 119901 (in C) rangingfrom costs of rawmaterials energy amortization labor forcemaintenance and taxes to direct and indirect costs119873119901(119905) is afunction accounting for the number of products carried outper hour It is calculated like

119873119901 (119905) =119870

119905(119878119896)120583 (24)

where 119878119896 is the set of tasks needed to manufacture andassemble a certain product (119901) and it constitutes the workload where 119896 represents the number of tasks 119905(119878119896) = sum119895isin119878119896 119905119895is the cumulated task time and it is called the product time Acubic function of 119905min has been considered 120583 is a parameterthat refers to the economic environment and the marketseasonality 119870 is a constant related to the current number ofworking hours per year

Each one of these tasks is performed by the robot armwhich uses a certain time to describe the optimal trajectoryAs above mentioned the developed algorithm (Section 3)returns the minimum time 119905min119901 to perform the task of therobot arm in order to obtain the product119901 while consideringthe time of the other tasks as constant The lower the timeused by the robot to perform its task the greater the numberof products manufactured per hour Then the cumulativetime of all tasks can be defined as follows

119905 (119878119896) = 119905min119901 +119896

sum119895notin119878robot

119905119895 (25)

Besides the amount that an additional item adds to acompanyrsquos total revenue during a period is called themarginalrevenue of the product (MRP)

This factor is defined as the additional products man-ufactured per hour because of reducing the time used bythe robot arm (119905min119901)The additional products manufacturedincrease the companyrsquos output and therefore the companyrsquostotal revenue

The marginal revenue product can be obtained by mul-tiplying the marginal product (MP) of the factor by themarginal revenue (MR) In a perfectly competitive marketthe marginal revenue a company receives equals the market-determined price of the product 119875119901

Therefore for companies in perfect competition themarginal revenue product MRP can be expressed as follows

MRP = MP times 119875119901 (26)

The law of diminishing marginal returns tells us that ifthe quantity of a factor is increased while other inputs areheld constant its marginal product will eventually decline Ifmarginal product is falling (MP darr) MRP must be falling aswell (MRP darr)

The marginal revenue of a product (calculated inSection 7)will be used to obtain the total annual benefits in anassembly line as well as to determine the trade-offs betweenthe benefits and the times obtained in the multiobjectiveoptimization problem

6 Pareto Optimality

Many real-world problems face two different types of mathe-matical difficultiesThose difficulties are the existence ofmul-tiple and conflicting objectives and a highly complex searchspace Contrary to a single optimal solution competing goalsentail a set of compromise solutions generally denoted bythe Pareto-optimal for example [18] When there is lack ofpreference information none of the corresponding trade-offsbetween decision variables could be said to be better thanthat of others The optimal set of solutions in multiobjectiveoptimization problems is named the Pareto-optimal set

A solution is defined as Pareto optimal if no improvementin one objective can be accomplished without adverselyaffecting at least one other objective In the objective spacethe hypersurface that represents all possible Pareto-optimalsolutions is termed as the Pareto front or frontier A designthat is located along the Pareto front is neither better norworse than any other solution along the Pareto front Hencethe solutions that compose the Pareto-optimal set are equiv-alently optimal The objective of multiobjective optimizationusing this technique is to generate as many Pareto-optimalsolutions as possible to adequately represent the Pareto frontThis allows obtaining sufficient information for a trade-offdecision between competing variables The Pareto front canbe discontinuous concave or convex and in general is notknown a priori

The concept of domination enables the comparison ofa set of designs with multiple objectives Such a concept isnot required for single objective optimization on account ofthe fact that the value of the objective function is the onlymeasure of the quality of the design

Then in a direct comparison of two designs if onedesign dominates another the dominating design is superiorand nearer to the Pareto front Instead if neither designdominates the other the designs are nondominant to eachotherTherefore the best designs (with equally good objectivevectors) in an arbitrary set of solutions can be distinguishedbecause they are not dominated by any other design inthe set they compose the nondominated subset Similarlythe designs that compose the Pareto-optimal set are thenondominated set associated with the entire feasible spaceand are located along the Pareto front


Table 2 Working times needed to perform the trajectory for each example

Optimizationalgorithm

Example 1 2 3 4 5 6 7 8 9 10Working time (s) 07488 05909 07511 15557 07488 05115 13920 07118 04529 10000


SimulationsoftwareGRASP



Consequently the trajectory planning optimization prob-lem in robotics can be defined as finding a motion law alonga given geometric path taking into account some predefinedrequirements while generating minimum trajectory time ofthe robot armThe inputs of the trajectory planning problemare the geometric path and the kinematic and dynamicconstraints while the output is the trajectory of the joints(or of the end-effector) The trajectory is expressed as a timesequence of position velocity and acceleration values Theoptimized trajectory should alsomeet the physical limitationsof the robot the constraint of energy consumption andcollisions avoidance

The times obtained from the simulation-optimizationprocedures lead to different benefits Therefore the Paretofronts can be determined thus showing the trade-offsbetween the benefits and the obtained times They showthe opportunity cost of time higher than the minimum (seeSection 7)

7 Results of the Application of theMethodology to Different Examples

The proposed multiobjective optimization methodology andthe Pareto optimality have been applied to different examplesin order to set the above mentioned trade-offs betweenthe benefits and the obtained times Therefore now theobjective is conducting an economic study to quantify theproductivity of the assembly line by comparing theminimumtime trajectory to perform certain industrial tasks It isobtained using two methods an optimization algorithm andthe simulation software GRASP

These times are obtained while taking into considerationthe physical constraints of the real working problem and theeconomic issues involved in the process Twenty exampleshave been solved The examples differ in the initial and finalconfigurations of the robot that is the location of the end-effector (see Table 2) The optimization algorithm simulatesthe PUMA 560 robot

This industrial robot arm is probably the most commonrobot in university laboratories It is a 6-R (revolution)type robot with 6 degrees of freedom and uses directcurrent (DC) servo motors as its actuators This robot isa compact computer-controlled robot not only to performservice tasks but also to carry out medium-to-lightweightassembly welding materials handling packaging inspection

applications personal care and so forth The Series 500 isthe most widely used model in the PUMA line of electricallydriven robots

With a 36-inch reach and 5-pound payload capacitythis robot is designed with a high degree of flexibilityand reliability The range of these angles from 1205791 to 1205796 isthe following (320∘ 250∘ 270∘ 280∘ 200∘ and 532∘) Thecorresponding link lengths from L1 to L6 are (432 432 43356 56 and 375) mm

Table 2 presents the results obtained for the developedalgorithm and GRASP that is the working time required forthe robot to perform the industrial tasks

With regard to the economic issues associated withthe robot industrial tasks we suppose for all examples thefollowing quantities and considerations

unitary cost to produce a certain item 08 C (withoutconsidering the cost of the energy consumed)

item price 1 C For the sake of simplicity we assumethat only one product is produced at this point

When the cost of the energy consumed is consideredthe different examples have different costs because of thedifferent working times Therefore cost of the consumedenergy 00676 CkWh (it is an average cost) This cost hasbeen added to the cost of 08 C

For reasons of clarity the manufactured products areobtained in only one shift of 8 hours (365 working days ina year) The benefits 119861 are presented for a period of one year

The time of the other industrial tasks needed to producethe item has been defined as 90 s that is the summation oftimes shown in (25) sum119896119895notin119878robot 119905119895

The optimization algorithm presents different workingtimes for the different examples thus leading also to differentbenefits

For instance the case number 19 which has no con-straints in both the jerk and the energy consumed presentsthe maximum annual revenue (Figure 1) Contrary case4 with severe physical constraints shows the minimumbenefits

The benefits for all examples obtained by means of theoptimization algorithm are depicted in Figure 1 With thecurrent demand the mean value of the benefit for the 20examples analyzed is 2314240 Cyear while the standarddeviation is 9032


2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)

Figure 1 Annual revenue for each example based on the currentdemand

The industrial tasks are carried out in only a few secondsso that the time scale is based on seconds Note that theGRASP working times are similar to those obtained with thedeveloped optimization algorithm although slightly higher

Then total benefits are also similar to those reported inFigure 1 although the higher working times lead to lowerbenefits

Consequently a lower mean value of the benefits isobtained that is 2313284 Cyear with a standard deviationof 9602

Note that different prices costs and number of itemsproduced by year may lead to higher differences in benefitsbetween the optimization algorithm and GRASP Howeverthe selected values are only intended to illustrate the impor-tance of using efficient algorithms of robot trajectory opti-mization for saving time and reducing costs in productionlines

In addition a new analysis is carried out using Paretooptimality to illustrate the loss of benefits on account of notusing optimization algorithms

For that we consider that three different products canbe manufactured and assembled in the same productionline The loss of benefits is represented by the Pareto frontsfor three different products They differ in their cumulativetime to be manufactured and assembled but share the sameworking time (119905min119901) of the robot arm

Then the minimum trajectory time for case 4 is used forthe three products in this example 155 sThese products alsodiffer in the total costs (without considering the energy costs)prices and values of the parameter 120583 which is intendedto simulate different economic environments and marketseasonality The total cost of Product 1 is 08 C Product 2 is082 C and Product 3 is 084 C while the prices are 10 C forProduct 1 105 C for Product 2 and 103 C for Product 3 Inthis analysis 119905(119878119896) has been defined as a cubic function of119905minp The parameter 120583 takes the values for each product of06 05 and 055 respectively

Consequently if the market conditions do not changeand the optimization algorithm is not used the minimumtrajectory time is not obtained

In this scenario there is a benefit loss due to the fact thatrobot arm may present higher working times

Moreover the multiobjective optimization problemallows obtaining the Pareto frontiers which provides

0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536

Execution time (s) for case 4Product 1Product 2Product 3

Bene

fits (

Cye

ar)

Figure 2 Pareto frontiers obtained with the optimization algorithmfor ldquoexample 4rdquo and the three different products manufactured

information about the trade-off of the decision variablesOne main trade-off is between the benefits and the workingtime (ie the Pareto frontier) Results for example 4 areshown in Figure 2 considering the manufacturing of threedifferent products That is the algorithm allows quantifyingthe benefit loss because of not using the optimizationalgorithm Each solution in the front will have optimalobjective function value optimal value of variables and theconstraints value All constraints will be satisfied by anysolution in the Pareto optimal front

It is worth pointing out that for the analyzed examplesthe differences between their annual energy costs are almostnegligible compared with the other costs

Furthermore the concept of nondomination sorting canbe used to categorize each design within a set into a hierarchyof nondominated levels or fronts Each different level ofnondomination represents a relative distance from the Paretofront The best nondominated front is closest to the Paretofront and each subsequent front lags further behind and ishence increasingly inferior

Through this sorting each design is associated witha front that defines the quality of the design relative tothe rest of the group To isolate the various fronts thedesigns that belong to the nondominated subset of the entiregroup are first identified These designs are the best in thegroup the closest to (or members of) the Pareto frontFor instance in our multiobjective trajectory optimizationthe nondominated subset (ie the best solution in termsof greater benefits) is represented by the Pareto frontier ofProduct 2 for times higher than 196 s (see Figure 2)

Any design belonging to this front is then temporarilyset aside and another comparison process determines thenext level of nondominated designs from the remainingpopulation

This nondominated subset is the front of Product 3 andthe procedure is repeated until the entire population has beensorted into the appropriate level that is the Pareto frontier ofProduct 1

Note however that for working times lower than 176 sthe Pareto frontier of Product 1 dominates the frontier ofProduct 2That is higher benefits are expected to be achievedin the assembly line for Product 1


020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098

Cost fluctuation (case 4)

Costs (C)

Bene

fits (

Cye

ar)

Figure 3 Benefits obtained for Product 1 and ldquoexample 4rdquo versuscosts

020000400006000080000

100000120000140000

083 096 105 121 132 133 153

Price fluctuation (case 4)

Price (C)

Bene

fits (

Cye

ar)

Figure 4 Benefits obtained for Product 1 and ldquoexample 4rdquo versusprice

Additionally for the twenty examples analyzed in thiswork a normal distribution function of the cost (119862) and theprice (119875) has been defined for a certain product based on thecurrent market economic fluctuations The benefits resultingfrom these market fluctuations are provided in Figures 3 and4

Figure 3 has been obtained when a normal distribution ofthe cost fluctuation is considered on account of hypotheticalmarket changesThe statistics that define the normal distribu-tion (mean and variance) are based on the current total costsabove mentioned

Figure 4 has been obtained when a normal distribution ofthe price fluctuation is considered on account of hypotheticalmarket changes The statistics that define the normal distri-bution (mean and variance) are based on the current prices

For the sake of conciseness the cost and price fluctuationshave been considered only for Product 1 Its current marketvalues are 119898119888 = 08 C for costs and 119898119901 = 1 C for priceswhile the standard deviation (120590) is defined as1198983With thesefunctions the market changes regarding costs and pricesare intended to be modeled 119862(119862 = 119862119901 lowast 119873119901) gives thecost to manufacture 119901 products (119873119901) 119875119901 gives the revenue asdefined by (2) that is multiplying the price of the productsmanufactured

These functions are sampled with these values being usedin the multiobjective optimization problem to obtain theresults presented in Figures 3 and 4

These figures show a hypothetic fluctuation in the marketconditions with regard to costs and prices for case 4

The fluctuations can be directly translated into benefitsusing the algorithm thus allowing managers in the decisionmaking process regarding which products should be manu-factured and also to define an efficient schedulingThereforethe design and planning of the robot tasks are considerablyimproved

Finally the algorithm has also been run for example 4 tosimulate an increase in the future demand of a 20 comparedwith the current demand (ie a total of 137778 productsmanufactured per year) The aim is to answer the questionabout how many extra working hours are needed to respondto that increase in the demand The best solution is givenby the optimization algorithm since it reports the minimumtrajectory time The solution is that we need an additional3504 hours per year to meet such demand This informationcan be used during the decision making process to design anefficient scheduling

8 Conclusions

This work deals with trajectory planning of industrial robotsfor assembly lines in a cost-efficient way thus overcom-ing limitations of the economic analysis methods whichare currently available It has been demonstrated that themultiobjective optimization algorithm finds the minimumtime trajectory of industrial robots and the maximum annualrevenue This means greater annual revenue and betteradaptation to market fluctuations in terms of costs pricesand product demands This is carried out by taking underconsideration the physical constraints of the real workingproblem and the economic issues involved in the processThe proposed procedure has been successfully validated indifferent examples of robotic industrial tasks where a betterplanning and design of production lines have been found

The results from different examples have been comparedusing two methodologies an optimization procedure and asimulation technique

We have checked that the results obtained with theoptimization procedure lead to lower working times andtherefore greater annual revenues in comparison with thoseobtained with the simulation technique Consequently thenumber of products manufactured and profits are increasedwhile the number of the shifts required is reducedThe core ofthis paper is the procedure to obtain the best working times inreal complex industrial robots with many degrees of freedomand mechanical constraints

This has shown the worth of the methodology with theoverall objective of improving the profitability of productionlines by designing flexible manufacturing systems Further-more an entire set of equally optimal solutions for eachprocess the Pareto-optimal sets are generated

This provides information about the trade-offs betweenthe different competing variables of the multiobjectiveoptimization problem (ie working times and profits forthe different products that can be manufactured at theproduction line)


Once the optimal time to perform each process isobtained in a cost-effective manner the results can be usedfor improving a wide variety of robotic industrial tasks Thiscan help managers in the decision making process regardingwhich products should be manufactured and to define anefficient scheduling to produce themThis is because it allowsadjusting the number of shifts needed according to the exist-ing demand of the products manufactured Then companiesmay stay competitive because the algorithm allows a quickadaptation to rapidly changing markets

As a further research this methodology will be extendedto deal with new decision variables in the multiobjectiveoptimization problem such as the energy consumed and timesimultaneously since they are conflicting variables

Conflict of Interests

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

Acknowledgment

This paper has been possible due to the funding from theScience and InnovationMinistry of the Spanish Governmentby means of the Researching and Technologic DevelopmentProject DPI2010-20814-C02-01 (IDEMOV)

References

[1] J E Bobrow SDubowsky and J S Gibson ldquoTime-optimal con-trol of robotic manipulators along specied pathsrdquo InternationalJournal of Robotics Research vol 4 no 3 pp 3ndash17 1985

[2] K G Shin and N D McKay ldquoMinimum -time control ofrobotic manipulators with geometric path constraintsrdquo IEEETransactions on Automatic Control vol AC-30 no 6 pp 531ndash541 1985

[3] Y Chen and A A Desrochers ldquoStructure of minimum-timecontrol law for robotic manipulators with constrained pathsrdquoin Proceedings of the IEEE International Conference on Roboticsand Automation pp 971ndash976 Scottsdale Ariz USA May 1989

[4] J Mattmuller and D Gisler ldquoCalculating a near time-optimaljerk-constrained trajectory along a specified smooth pathrdquoInternational Journal of Advanced Manufacturing Technologyvol 45 no 9-10 pp 1007ndash1016 2009

[5] K J Kyriakopoulos and G N Saridis ldquoMinimum jerk pathgenerationrdquo in Proceedings of the IEEE International Conferenceon Robotics and Automation vol 1 pp 364ndash369 PhiladelphiaPa USA 1988

[6] D Simon and C Isık ldquoA trigonometric trajectory generator forrobotic armsrdquo International Journal of Control vol 57 no 3 pp505ndash517 1993

[7] D Constantinescu and E A Croft ldquoSmooth and time-optimaltrajectory planning for industrial manipulators along specifiedpathsrdquo Journal of Robotic Systems vol 17 no 5 pp 233ndash2492000

[8] P Huang Y Xu and B Liang ldquoGlobal minimum-jerk trajectoryplanning of spacemanipulatorrdquo International Journal of ControlAutomation and Systems vol 4 no 4 pp 405ndash413 2006

[9] D Garg and C Ruengcharungpong ldquoForce balance and energyoptimization in cooperatingmanipulatorsrdquo inProceedings of the

23rd Annual Pittsburgh Modeling and Simulation Conferencepp 2017ndash2024 Pittsburgh Pa USA 1992

[10] A R Hirakawa and A Kawamura ldquoProposal of trajectory gen-eration for redundant manipulators using variational approachapplied to minimization of consumed electrical energyrdquo inProceedings of the 4th International Workshop on AdvancedMotion Control (AMC rsquo96) pp 687ndash692 Mie Japan March1996

[11] B H Cho B S Choi and J M Lee ldquoTime-optimal trajec-tory planning for a robot system under torque and impulseconstraintsrdquo International Journal of Control Automation andSystems vol 4 no 1 pp 10ndash16 2006

[12] G Field and Y Stepanenko ldquoIterative dynamic programmingan approach tominimumenergy trajectory planning for roboticmanipulatorsrdquo in Proceedings of the 13th IEEE InternationalConference on Robotics and Automation pp 2755ndash2760 Min-neapolis Minn USA April 1996

[13] X F Zha ldquoOptimal pose trajectory planning for robot manipu-latorsrdquoMechanism andMachineTheory vol 37 no 10 pp 1063ndash1086 2002

[14] F Valero V Mata and A Besa ldquoTrajectory planning inworkspaces with obstacles taking into account the dynamicrobot behaviourrdquo Mechanism and Machine Theory vol 41 no5 pp 525ndash536 2006

[15] F J Valero J L Suner V Mata and F J Rubio ldquoOptimal timetrajectory for robots with torque jerk and energy constraintsrdquoin Proceedings of the ECCOMASThematic Conference on Multi-body Dynamics 2009

[16] F Rubio F Valero J Sunyer and V Mata ldquoDirect step-by-stepmethod for industrial robot path planningrdquo Industrial Robotvol 36 no 6 pp 594ndash607 2009

[17] J L Suner F J Valero J J Rodenas and A Besa ldquoComparacionentre procedimientos de solucion de la interpolacion porfunciones splines para la planificacion de trayectorias de robotsindustrialesrdquo in Proceedings of the 8th Congreso Iberoamericanode Ingenierıa Mecanica Cusco Peru October 2007

[18] R Sanders ldquoThe pareto principle its use and abuserdquo Journal ofProduct amp Brand Management vol 1 no 2 pp 37ndash40 1992

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


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


of assembly lines Finally the conclusions will be discussed inthe last section

2 Background of the TrajectoryPlanning Problem

Currently there are a great number of methodologies to solvethe trajectory planning problem for industrial robots whichgive the time needed to perform a task But few papers tacklethe analysis of productivity related to the working timesobtained

Over the years the algorithms have been polished andthe working assumptions of the robotic systems have beenincreasingly adjusted to real conditions This fact has beenachieved by analysing the complete behaviour of the roboticsystem particularly the characteristics of the actuators andthe mechanical structure of the robot To tackle this problemother important working parameters and variables have beentaken into account such as the input torques the energyconsumed and the power transmitted Furthermore thekinematic properties of the robotrsquos links such as the veloc-ities accelerations and jerks must be also considered Theaforementioned algorithms provide a smooth robot motionfor the robotic system

To obtain the best trajectories in terms of minimumtimes some of the working parameters have been includedin the appropriate objective function of the optimizationprocedure (input torques the energy consumed and thepower transmitted) The optimization criteria most widelyused can be sorted as follows

(1) Minimum time required which is directly boundedto productivity

(2) Minimum jerk which is bounded to the quality ofwork accuracy and equipment maintenance

(3) Minimum energy consumed or minimum actuatoreffort both linked to savings

(4) Hybrid criteria for example minimum time andenergy

In the past the early algorithms that solved the trajectoryplanning problem tried to minimize the time needed forperforming the task (see [1ndash3]) One disadvantage of thoseminimum-time algorithms was that the trajectories haddiscontinuous values of acceleration and torques which led todynamic problems during the trajectory performance Thoseproblems were avoided by imposing smooth trajectories tobe followed such as spline functions which have been usedin both path and trajectory planning

The early algorithms in trajectory planning sought tominimize the time needed for performing the task Thedynamics properties of actuators were neglected A recentexample of this type of algorithm can be found in [4] whichdetermines smooth and near time-optimal path-constrainedtrajectories It considers not only velocity and acceleration butalso jerk

Later the trajectory planning problem was tackled bysearching for jerk-optimal trajectories Jerks are highly

important for working with precision and without vibrationThey also have an effect on the control system and thewearingof mobile parts such as joints and bars These methods allowa reduction in errors during trajectory tracking the stressesin the actuators and also in the mechanical structure ofthe robot and the excitement of resonance frequencies Jerkrestriction is introduced by other authors [5 6]

In [7] a method is introduced for determining smoothand time-optimal path-constrained trajectories for roboticmanipulators by imposing limits on the actuator jerks

In [8] a global minimum-jerk trajectory planning algo-rithm of a space manipulator is presented

Another different approach to solving the trajectoryplanning problem is based on minimizing the torque andthe energy consumed instead of the trajectory time or thejerk This approach leads to smoother trajectories An earlyexample is seen in [9]

Similarly in [10] the authors searched for the minimumenergy consumed They proposed a method for solvingthe trajectory generation problem in redundant degree offreedommanipulatorsThey used a variational approach andthe B-Spline curve was introduced to minimize the electricalenergy consumed in a robot manipulator system

Thework in [11] also takes into account energyminimiza-tion for the trajectory planning problem

In [12] the authors proposed a technique of iterativedynamic programming to plan minimum energy consump-tion trajectories for robotic manipulators The dynamicprogramming method was modified to perform a series ofdynamic programming passes over a small reconfigurablegrid covering only a portion of the solution space at anyone pass Although strictly no longer a global optimizationprocess this iterative approach retained the ability to avoidcertain poor local minima while avoiding the dimensionalissue associatedwith a pure dynamic programming approachThe modified dynamic programming approach was veri-fied experimentally by planning and executing a minimumenergy consumed path for a Reis V15 industrial manipulator

Afterwards new perspectives appear for solving thetrajectory planning problem The main point was to use aweighted objective function to optimize the working param-eters [13] There the cost function is a weighted balance oftransfer time the mean average of the torques and power

In this paper we will introduce two methods to solve thetrajectory planning problem for industrial robots workingin complex environments The time will be used in theeconomical study

In the first method the procedure calculates the optimaltrajectory by neglecting initially the potential presence ofobstacles in the workspace By removing the obstacles (realor potential) from the optimization problem the algorithmwill calculate a minimum time trajectory as a starting pointThen the procedure must take into account the real obstaclespresented in the workspace When obstacles are consideredthe initial trajectory will not be feasible and will have toevolve so that it can become a solution The way this initialtrajectory evolves until a new feasible collision-free trajectoryis obtained is presented in this paper It is a direct algorithmthat works in a discrete space of trajectories approaching the






119895



119895

119909119894 120572119895

119910119894 120572119895




Find 119902 (119905) 120591 (119905) 119905119891 (1)



119905119891

0

119889119905 (2)




119872(119902 (119905)) 119902 (119905) + 119862 (119902 (119905) 119902 (119905)) 119902 (119905) + 119892 (119902 (119905)) = 120591 (119905) (3)





(4)


119902 (0) = 119902119900 119902 (119905119891) = 119902119891

119902 (0) = 0 119902 (119905119891) = 0

(5)


119889119894119895 ge 119903119895 + 119908119894 (6)



120591min119894 le 120591119894 (119905) le 120591



119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(8)


119902min119894 le

119902119894 (119905) le



(9)

(8) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (10)







dofsum119894=1

(119902119897119894 minus 119902119896119894 )2 (11)






(12)




119902119894119895 (119905119895minus1) = 119902119895

119894

119902119894119895 (119905119895) = 119902119895+1

119894 (13)


1199021198941 (1199050) = 0

119902119894119898minus1 (119905119898minus1) = 0(14)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (15)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (16)


120591min119894 le 120591119894 (119905) le 120591



119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(18)


119902min119894 le

119902119894 (119905) le


(j) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (20)



119898minus1

sum119895=1

Δ119905119895 = 119905min (21)


(a) 119862119896 = 119862119895 cup 119888119899(b) 119899 = 1(c) 119899 = 119898 + 1






















120596max =119875

119905min (22)












Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)


119873119901 (119905) =119870

119905(119878119896)120583 (24)



119905 (119878119896) = 119905min119901 +119896


119905119895 (25)





MRP = MP times 119875119901 (26)



6 Pareto Optimality
































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119895



119895

119909119894 120572119895

119910119894 120572119895




Find 119902 (119905) 120591 (119905) 119905119891 (1)



119905119891

0

119889119905 (2)




119872(119902 (119905)) 119902 (119905) + 119862 (119902 (119905) 119902 (119905)) 119902 (119905) + 119892 (119902 (119905)) = 120591 (119905) (3)





(4)


119902 (0) = 119902119900 119902 (119905119891) = 119902119891

119902 (0) = 0 119902 (119905119891) = 0

(5)


119889119894119895 ge 119903119895 + 119908119894 (6)



120591min119894 le 120591119894 (119905) le 120591



119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(8)


119902min119894 le

119902119894 (119905) le



(9)

(8) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (10)







dofsum119894=1

(119902119897119894 minus 119902119896119894 )2 (11)






(12)




119902119894119895 (119905119895minus1) = 119902119895

119894

119902119894119895 (119905119895) = 119902119895+1

119894 (13)


1199021198941 (1199050) = 0

119902119894119898minus1 (119905119898minus1) = 0(14)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (15)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (16)


120591min119894 le 120591119894 (119905) le 120591



119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(18)


119902min119894 le

119902119894 (119905) le


(j) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (20)



119898minus1

sum119895=1

Δ119905119895 = 119905min (21)


(a) 119862119896 = 119862119895 cup 119888119899(b) 119899 = 1(c) 119899 = 119898 + 1






















120596max =119875

119905min (22)












Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)


119873119901 (119905) =119870

119905(119878119896)120583 (24)



119905 (119878119896) = 119905min119901 +119896


119905119895 (25)





MRP = MP times 119875119901 (26)



6 Pareto Optimality
































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(12)




119902119894119895 (119905119895minus1) = 119902119895

119894

119902119894119895 (119905119895) = 119902119895+1

119894 (13)


1199021198941 (1199050) = 0

119902119894119898minus1 (119905119898minus1) = 0(14)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (15)


119902119894119895 (119905119895) = 119902119894119895+1 (119905119895) (16)


120591min119894 le 120591119894 (119905) le 120591



119875min119894 le 120591119894 (119905) 119902119894 (119905) le 119875


(18)


119902min119894 le

119902119894 (119905) le


(j) energy consumed

119898minus1

sum119895=1

(

dofsum119894=1

120576119894119895) le 119864 (20)



119898minus1

sum119895=1

Δ119905119895 = 119905min (21)


(a) 119862119896 = 119862119895 cup 119888119899(b) 119899 = 1(c) 119899 = 119898 + 1






















120596max =119875

119905min (22)












Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)


119873119901 (119905) =119870

119905(119878119896)120583 (24)



119905 (119878119896) = 119905min119901 +119896


119905119895 (25)





MRP = MP times 119875119901 (26)



6 Pareto Optimality
































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























120596max =119875

119905min (22)












Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)


119873119901 (119905) =119870

119905(119878119896)120583 (24)



119905 (119878119896) = 119905min119901 +119896


119905119895 (25)





MRP = MP times 119875119901 (26)



6 Pareto Optimality
































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Max119861 = 1

(1 + 119903)119879[

119899

sum119901=1

(119875119901 minus 119862119901) sdot 119873119901 (119905)] (23)


119873119901 (119905) =119870

119905(119878119896)120583 (24)



119905 (119878119896) = 119905min119901 +119896


119905119895 (25)





MRP = MP times 119875119901 (26)



6 Pareto Optimality
































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































2280022850229002295023000230502310023150232002325023300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Cases

Bene

fits (

Cye

ar)












0

5000

10000

15000

20000

25000

30000

156

176

196

216

236

256

276

296

316

336

356

376

396

416

436

456

476

496

516

536


Bene

fits (

Cye

ar)










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










020000400006000080000

100000120000140000

024 049 052 056 064 075 085 089 098


Costs (C)

Bene

fits (

Cye

ar)


020000400006000080000

100000120000140000

083 096 105 121 132 133 153


Price (C)

Bene

fits (

Cye

ar)










8 Conclusions











Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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