A simulated annealing heuristic for robotics assembly using the dynamic pick-and-place model

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A simulated annealing heuristic for robotics

assembly using the dynamic pick-and-place

model

Chao-Ton Su & Hsin-Pin Fu Published online: 15 Nov 2010.

To cite this article: Chao-Ton Su & Hsin-Pin Fu (1998) A simulated annealing heuristic for robotics assembly using the dynamic pick-and-place model, Production Planning & Control: The Management of Operations, 9:8, 795-802, DOI: 10.1080/095372898233560

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A

simulated annealing heuristic for robotics

assembly using the dynamic pick-and-place model

CHAO

-

TON SU

and

HSIN

-

PIN FU

Keywords simulated annealing, FPP, DPP, robotics as-sembly, sequence, assignment

Abstract. Products assembled by robots are typical in present day manufacturing.The traditional type of automatic assembly isFixedPicked andPlace (FPP) mode.The development of the

DynamicPicked-and-Place (DPP) model is an important issue in robotics travel. Until now, to route robotics travel, the authors usually have utilized the ® xed coordinate of insertion points and magazine of the Travelling Salesman Problems (TSP) method to sequence the insertion points. However, robotics travel routing should be based on a relative coordinate because the coordinates of insertion point and magazine are constantly changing.That is, the robotics, board and magazine are simultaneously moved at dierent speeds.This study pre-sents a Simulated Annealing (SA) -based algorithm that can arrange the insertion sequence and assign the magazine slots to obtain a performance better than in the traditional approach.

1. Introduction

The industrial robot has been applied widely in manu-facturing and is usually a high-production tool.In gen-eral, most products assembled by robots are electrical products high in unit value. Therefore, saved assembly cycle time being saved cost, it is important to reduce the assembly time to enhance productivity and competitive-ness.The most general assembly cells consist of the robot, assembly table ( board) and component slots ( magazine).

Three factors are highly correlated in their eects on overall assembly e ciency: ( 1) robot motion control; (2) the sequence for placing the individual component on the assembly board; and ( 3) the corresponding maga-zine slot assignment.To ® nd the optimal robot travelling routes is complicated and time consuming, especially

Authors: Chao-TonSu andHsin-PinFu,Department ofIndustrialEngineering andManagement,

NationalChiaoTungUniversity,Hsinchu,Taiwan.

Chao-TonSuis aProfessor in theDepartment ofIndustrialEngineering andManagement at

National ChiaoTung University, Taiwan. He received hisB.S. and M.S. from Chung Yuan

Christian University, Taiwan, and Ph.D. from University of Missouri-Columbia, USA, all in

IndustrialEngineering. His current research activities include quality engineering, production management and neural networks in industrial applications. Dr Su has published articles in Computers and Industrial Engineering, Computers in Industry, International Journal of Industrial Engineering, International J ournal of Production Research, Production Planning and Control, Integrated ManufacturingSystems,Opsearch,Quality andReliabilityEngineering International,InternationalJ ournal of Quality and Reliability Management,International Journal of Quality Science,Total Quality Management, Journal of theChineseInstitute of IndustrialEngineers.

Hsin-PinFuis a section chief atIndustrialDevelopmentBureau,Ministry ofEconomicAairs,

Taiwan.He holds aB.S. fromChungYuanChristianUniversity,Taiwan, anM.S. fromUniversity ofMissouri-Columbia,USA and aPh.D. fromNational ChiaoTungUniversity, Taiwan, all in

IndustrialEngineering. His current research interests are in operation management and local search methods in industrial applications.Dr Fu has published articles inInternational Journal of IndustrialEngineeringandIntegratedManufacturingSystems.

0953-7287/98 $12.00 Ñ 1998 Taylor & Francis Ltd.

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when many components need to be assembled. In prac-tice, the heuristic solution is highly desirable.

Two types of robot assembly problems have been char-acterized based on dierent robot motions.They are: (1) ® xed robot motion between ® xed pick and place (FPP) points; and (2) robot motion with dynamic pick and place (DPP) points.In theFPPmotion model, the maga-zine ( or component slots) moves horizontally along anX -axis and the robot moves only vertically along a Y-axis. The assembly board (X

-

Y table) moves freely in any direction so that the magazine can move required com-ponents to the ® xed pick-up points.When the assembly board moves to a ® xed placement location, the robot picks up and places the components among these two ® xed points. Figure 1 shows the layout of the FPP

approach. Few researchers have developed assembly sequence methods, instead focusing on the FPP mode (Randhawa et al. 1985, Cunnigham andBrowne 1986,

Ball and Magazine 1988, Mettalla and Egbelu 1989,

Egbelu et al. 1996).Because theFPP approach involves undesirable robot waiting time at the ® xed pick-up and placement points,Su et al. ( 1995) developed robot moves with a ¯ exibleDPPapproach using a heuristic method to eliminate the robot waiting time. Su et al. ( 1995) also show that the DPP approach is superior to the FPP

approach in most cases where magazine slots are assigned randomly.

To obtain the shortest robot travel routing, the as-sembly sequence and magazine assignment are relatively important.The better the assembly sequence and maga-zine assignment, the shorter the moves of the assembly table and magazine. Two issues are thus involved: (1) how to arrange the insertion ( assembly) sequence; and (2) how to assign the corresponding components to spe-ci® c magazine slots. Su et al. ( 1995) dealt with robotics travel routing by the Travelling Salesman Problems (TSP) method (Karg and Thompson 1964) based on the ® xed insertion point coordinates and random maga-zine assignments. Wang et al. ( 1997) indicated that rea-sonable allocation of the magazine slots instead of random assignment improves performance, and he devel-oped a heuristic magazine assignment approach to

opti-mize theDPP method. The assembly cell of one board and one magazine ( 1B1M) is applied in Su andWang’s approaches.Wang ( 1996) also developed some layouts of assembly cell, e.g. 1B2M and 2B1M based on theDPP

mode.Nevertheless,Wang’s approach was still based on a ® xed coordinate using the TSP method to obtain robotics travel routing. In fact, robotics travel routing should be based on relative coordinates not ® xed coordi-nates because the coordicoordi-nates of the insertion point and magazine change at all times, i.e. the robotics, board and magazine are simultaneously moved at dierent speeds.

Su and Wang’s approaches did not consider the simul-taneous movement of robotics, board and magazine, and how this in¯ uences coordinates solving all the time dur-ing the assembly. Therefore, their approaches are not suitable for solving the robotics assembly problem. In this study, aSimulatedAnnealing (SA) -based algorithm is presented to obtain the shortest cycle time, arrange the insertion ( assembly) sequence, and assign the correspond-ing components to speci® c magazine slots based on the

DPProbot motion model.Simulation results demonstrate that the proposed approach can signi® cantly reduce the assembly cycle time.

2. DPP background

In theDPP model, the assembly board and magazine move only horizontally along theX-axis; the robot moves vertically along theY-axis, and the pick-up and place-ment points are dynamically allocated. Figure 2 shows the layout of theDPPmodel.

To describe theDPPmodel more clearly, the following notations are given:

CT the cycle time to assemble all components N the number of insertion locations K the number of component types

m(i) the magazine pick-up location of the i-th assembly sequence

b(i) the placement location of the i-th assem-bly sequence

796 Chao-TonSu andHsin-PinFu

Figure 1. The layout of theFPPapproach.

Figure2. The layout of theDPPmode.

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T R(b(i)

,

m(i)) robot travel time from board location b(i)

to magazine location m(i)

T R(m(i)

,

b(i)) robot travel time from magazine location

m(i) to board location b(i)

Vr the average speed of the robot

Vb the average speed of the assembly board Vm the average speed of the magazine T P the time needed to pick up a component

upon arrival

T I the time needed to insert a component upon arrival

A(xi

,

yi) the coordinate of x and y at pointA P Q the distance between pointsP andQ

We assumed that the initial location of the robot is at the right upper coordinate of the ® rst component pick-up point. Also, the initial location of board and magazine are at the x coordinate of the ® rst insertion point ( see ® gure 3). The robot returns to the initial location upon completion of the assembly process, ready to assemble the next product.

Figure 4 shows the basic layout of theDPPmodel.The insertion placement point is decided as follows: when the robot picks up the i-th component at point D(xi

,

yi) on

the magazine and then moves to the insertion point

C(xi

,

yi), two insertion points are possible [® gure 4(b) ]

due to the limitations of robot speed, board speed and the board’s insertion point. It is assumed that point A(xi

,

yi) and point B(xi

,

yi) are two possible placement

locations, and point C(xi

,

yi) is the relative coordinate

location of the i-th component in the insertion sequence immediately after the robot ® nishes inserting the ( i

-

1)-th component.The coordinate of the pointD(xi

,

yi)

loca-tion should be determined during the preceding insertion.The placement location A(xi

,

yi) is used when

the robot reaches that point from pointD(xi

,

yi) after the

board arrives at point A(xi

,

yi) from point C(xi

,

yi). In

other words, insertion takes place at point A(xi

,

yi) if

the following is true:

T R(b(i

-

1)

,

m(i))+T P +A D /Vr

³

CA/Vb (1) where T R(b(i

-

1)

,

m(i)) is determined during the pre-ceding insertion operation and A D/Vr =T R(m(i)

,

b(i)).

The i-th insertion point moves from point C(xi

,

yi) to

point A(xi

,

yi) and waits for the robot, which travels in

the y direction for a distance ofA D only.Then, the pla-cement coordinate location at pointC(xi

,

yi) is set by

C(xi) =D(xi) and C(yi) =A(yi) (2) Otherwise, when the robot reaches point A(xi

,

yi) from

point D(xi

,

yi) before the board arrives at pointA(xi

,

yi)

from pointC(xi

,

yi), the possible interception of the board

movement by the robot occurs at pointB(xi

,

yi).That is,

the robot intercepts the i-th insertion point A(xi

,

yi), at

pointB(xi

,

yi) and the following relation holds:

T R(b(i

-

1)

,

m(i))+T P +D B /Vr=CB /Vb (3) Equation ( 3) can also be expressed as

Figure 3. The initial setup location.

Figure 4. Possible movement of theDPPmodel.

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T R(b(i

-

1)

,

m(i))+T P

+

[

(D(xi)

-

B(xi))2+(D(yi)

-

B(yi))2

]

1/2 Vr

=

|

B (xi)

-

C(xi)

|

Vb (4)

Also, theB(yi) placement interception point by the robot

at pointB(xi

,

yi) is set byB(yi) =A(yi).

In the same situation, to decide the pick-up coordinate location on a magazine is also necessary.It is similar to determining the placement locations on a board. Both pointsA(xi

,

yi) andB (xi

,

yi) are also possible pick-up

loca-tions, and C(xi

,

yi) is the relative coordinate location of

the i-th insertion point in the pick-up sequence immedi-ately after the robot ® nishes picking up the (i

-

1)-th component [® gure 4(a) ]. The point D(xi

,

yi) location

has also been determined during the preceding insertion.

The pick-up location A(xi

,

yi) is used when the robot

reaches pointA(xi

,

yi) from pointD(xi

,

yi) after the

maga-zine arrives at pointA(xi

,

yi) from pointC(xi

,

yi).In other

words, the pick-up takes place at point A(xi

,

yi) if the

following is true:

T R(m(i

-

1)

,

b(i))+T I +D A/Vr

³

CA/Vm (5)

the magazine indexes the i-th pick-up point,C(xi

,

yi), to

point A(xi

,

yi).The pick-up coordinate location at point A(xi

,

yi) is given by

A(xi) =D(xi-1) and A(yi) =C(yi) (6) Otherwise, if the robot reaches pointA(xi

,

yi) from point D(xi

,

yi) before the magazine arrives at point A(xi

,

yi)

from point C(xi

,

yi), then the robot intercepts to pick up

the component at pointB (xi

,

yi) and the following

rela-tional equation ( 7) holds:

T R(b(i

-

1)

,

m(i))+T I +D B/Vr =CB /Vm (7)

where T R(b(i(1)

,

m(i)) is known and D B/Vr =

T R(b(i

-

1)

,

m(i)).Equation ( 7) can also be expressed as T R(b(i

-

1)

,

m(i))+T I

+

[

(D(xi)

-

B(xi))2+(D(yi)

-

B(yi))2

]

1/2 Vr

=

|

B (xi)

-

C(xi)

|

Vb (8)

and then the robot± magazine movement interception point is setB(yi) =C(yi).

Equations ( 1), ( 3), ( 5) and ( 7) described above express both cases whether robot interception happens or not, while the robot begins its movement from the pick-up coordinate location to the place coordinate location D(xi

,

yi), simultaneously the magazine begins to index

the proper component type to pointA(xi

,

yi) orB(xi

,

yi). In this study, we used the assembly cycle time (CT) to evaluate the assembly e ciency.The shorter the assem-bly cycle time, the better the assemassem-bly e ciency.

Equation ( 9) expresses totalCT ( not includingT P and T I) as a function of the total robot travelling distance divided by robot speed.IfVmandVb are fast enough so that the assembly table and magazine can move to the points before the robot arrives, no robot interception of a moving board or magazine occurs in this optimal case.

We achieve optimal assembly cycle time when equations ( 1) and ( 5) are true, and then the total cycle time in equation ( 9) also should be optimal.

CT = Ni=1T R(m(i)

,

b(i))+ Ni=1T R(b(i)

,

m(i +1))

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where m(N + 1) =m(N).

In fact, equations ( 1) and/or ( 5) may not hold in any case due to the speed limitation ofVm andVb.To avoid the robot idling at A(xi

,

yi), robot interception then

occurs. Equations ( 3) and (7) represent the robot inter-ceptsA(xi

,

yi) atB(xi

,

yi) such that no robot waiting time

occurs, thus does theDPPmodel eliminate robot waiting time possible in theFPP model.

As mentioned above, the optimal condition ( shortest CT) happens when the robot travels only in theY direc-tion and no robot intercepdirec-tion occurs.In other words, the problem of increasing assembly e ciency can be con-verted to that of dealing with minimum overall robot interception distance.

3. Simulated annealing

The simulated annealing approach is a general combi-natorial optimization technique used to solve di cult problems through controlled randomization.SAis a glo-bal technique that attempts to avoid local optimization traps by allowing occasional increases of criteria, and emulates the annealing process which attempts to force a system to its lowest energy-controlled cooling.TheSA

annealing process can be described follows: ( 1) the tem-perature is raised to a su cient level; (2) the temperature is maintained at each level for su cient time; and ( 3) the temperature is allowed to cool under controlled con-ditions until the desired energy is reached.

The initial temperature, the amount of time the system remains at this temperature, and the cooling temperature rate are referred to as the annealing schedule. If the system is cooled too fast, it may `freeze’ at an undesirable, high energy level.The freeze of a system at an able energy state corresponds to the problem of undesir-able local optimization.

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TheSA algorithm requires that we de® ne: ( 1) a solu-tion’s con® guration; (2) an objective ( energy) function; ( 3) a generation mechanism; and ( 4) the annealing sche-dule. The most important issue in aSA algorithm is the annealing schedule consisting of: ( 1) the initial tempera-ture; (2) a cooling function; ( 3) the number of iterations to be performed at each temperature; and ( 4) a stopping criterion to terminate the algorithm.

The general procedure for implementing a simulated algorithm is as follows:

Step 1. Set an initial temperature,T, and an initial sol-ution X0. Let f0=f(X0) denoting the

corre-sponding objective value.Set i=0.

Step 2. Set i=i +1 and randomly generate a new sol-utionXi, and evaluate fi =f(Xi).

Step 3. D Z =fi

-

fi-1.IfD Z <0 ( downhill move) then

go to step 5.Otherwise ( uphill move) accept fi as

the new solution with probability e(-D Z/t).

Step 4. If fi was rejected in step 3 then set fi =fi-1.

Step 5. If the current objective value, fi, is satis® ed then

`freeze’. Otherwise, adjust the current tempera-tureT according to the annealing schedule and go to step2.

Detailed descriptions of theSAapproach and its applica-tions can be found in Rutrnbar ( 1989), Sridhar and

Rajendran ( 1993), Koulmas et al. ( 1994), Schmidt and

Jackmen ( 1995) , andChen et al. ( 1995) .

4. Simulated annealing in theDPP approach

To describe the proposed heuristic for theDPP model, the following notations are ® rst given:

T the initial temperature.

M the number of iterations for each temperature level.

CTi the i-th ® tness function ( ® nish assembly cycle

time)

K N the iteration number of temperature decreased N N the number of swap in insertion points and slots

on each iteration T T Ci the i-th optimal solution

R Pi the i-th energy probability in the i-th iteration A Pi the i-th random probability in the i-th iteration R the rate by which the temperature is decreased Xi the i-th iteration solution

fi(Pi

,

Si) the i-th solution in the insertion sequencePi

and magazine assignmentSi

D Z the dierence ofCTi

-

T T Ci-1

Next, the proposed procedure is presented in the fol-lowing:

Step1. SetT =100,I =0 andM =0, and set an initial solution X0=f0(P0

,

S0).

Step2. Set i=i +1.

IfT >5 thenN N =3.

IfT

³

0.1 andT

£

5 thenN N =2.

IfT <0.1 thenN N =1.

The insertion points and component slots swapped N N times, respectively, then obtain the new solution Xi=fi(Pi

,

Si).

Calculate theCTi ofXi. Step3. Set M =M + 1.

If M

³

30 then T =T

´

R(0.9), M =0 and K N = K N + 1.

Step4. IfCTi <T T Ci-1 then go to step 5,

else

D Z =CTi

-

T T Ci-1 andR Pi =e(-A Z/T). Select a probability forA Pi.

IfR Pi <A Pi thenT T Ci=T T Ci-1 and go to

step 6.

Step5. Set T T Ci =CTi andXi =fi(Pi

,

Si). Step6. IfK N

³

15 then `freeze’, else go to step2. 5. Simulation results

A numerical example fromWang et al. ( 1997) is pre-sented in this section which demonstrates the proposed heuristic’s eectiveness.Seven factors are selected in the study.Table 1 lists control factors and their experimental design levels. A total of 32 (25) combination runs ( table

2) are designed. For each combination, 30 sets of as-sembly locations and their component types are ran-domly generated by computer for simulation so that the averaged result is more objective.

For example, to obtain one set of robotics travel times in the case ofN assembly points andK component types, we decide robot, board and magazine speeds, and ran-domly generateN placement locations on the board and K corresponding component types. Based on the DPP

motion, to obtain one set of data takes about 5 min to run the program inBASIClanguage on a pentium-100

Table 1. Factors and their experimental design levels.

Factors Levels ( low/high)

Number of assembly points (N) 20/30

Number of component types (K) 10/15

Length of board (B L) 20/40 ( unit distance)

Width of board (B W) 15/25 ( unit distance)

Speed of robot (Vr) 6/12( unit distance/unit time)

Speed of board (Vb) 3/5.5 ( unit distance/unit time)

Speed of magazine (Vm) 2.5/4.5 ( unit distance/unit time)

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PC using the SA approach. Thus, the shortest robotics assembly cycle time, the insertion sequence, and the assignment of corresponding components to speci® c magazine slots can be determined. One combination is the average of 30 data sets obtained in the same way through theSA approach.

The swapped principle of insertion point and compon-ent slots in theSA approach is that the higher tempera-ture implies a greater swapping number. The initial temperature is set at 100ë C and progressively decreased according to the cooling schedule until frozen.If theCT of continuous 450 (M

´

K N) iterations is not decreased, then `freeze’ and a solution is obtained.Figure 5 shows an example of simulated annealing data (20 assembly points and 10 components).We can see how the annealing pro-cess works by the cooling schedule to freeze.

Simulation results are shown in ® gures 6± 9.Each unit on the abscissa for these ® gures stands for a combination in table 2. According to these ® gures, we ® nd that the performance of the SA approach is superior to Wang’s approach. The average assembly times of 32

combina-tions for the SA and Wang’s approach are shown in table 3. In the case of 30 assembly points and 15 com-ponent types, the average cycle times of theSAalgorithm and Wang’s algorithm are 78.46 time units and 88.71 time units, respectively. The reduction of average cycle time is 10.25 time units and the percentage of reduction is

Table2. Thirty-two combinaitns of ® ve factors.

B L B W Vr Vb Vm Combination 20 40 15 25 6 12 3 5.5 2.5 4.5 1 _* _* _* _* _* 2 _* _* _* _* _* 3 * * * * * * 4 * * * * * 5 * * * * * 6 * * * * * 7 * * * * * 8 * * * * * 9 * * * * * 10 * * * * * 11 * * * * * 12 _* _* _* _* _* 13 * * * * * 14 _* _* _* _* _* 15 * * * * * 16 _* _* _* _* _* 17 _* _* _* _* _* 18 * * * * * 19 _* _* _* _* _* 20 * * * * * 21 * * * * * 22 _* _* _* _* _* 23 * * * * * 24 * * * * * 25 * * * * * 26 * * * * * 27 * * * * * 28 * * * * * 29 * * * * * 30 * * * * * 31 * * * * * 32 _* _* _* _* _*

Figure 5. An example of freezing inSA(N = 20,K = 10) .

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11.56%. In other words, if there is a product where N =30 andK =15 and it is assembled by robotics, the production rate of the SA approach is 11.56% higher than Wang’s. This is a signi® cant improvement in terms of product due date or reduction of product cycle time. Moreover, ® gure 10 shows that the greater the insertion points and/or number of parts, the better the

performance.If a product has more insertion points and/ or component types than in the case of N =30 and K =15, it will enjoy a signi® cant advantage of more than 11.5%.

6. Conclusions

The robotics assembly problem is aNP-complete prob-lem.The development of theDPPmodel is an important

Figure 6. Simulation results for the case of20 assembly points and 10 component types.

Figure 7. Simulation results for the case of20 assembly points and 15 component types.

Figure 8. Simulation results for the case of 30 assembly points and 10 component types.

Figure 9. Simulation results for the case of 30 assembly points and 15 component types.

Figure 10. Acomparison of the reduction percentage between N andK.

Table 3. Acomparison of cycle times.

AverageC T AverageCT Percentage

ofWang’s ofSA C T of of

Cases approach algorithm reduction reduction

N =20, 52.25 49.67 2.58 4.94 K = 10 N =20, 55.07 50.70 4.37 7.93 K = 15 N = 30, 80.19 75.13 5.06 6.31 K = 10 N = 30, 88.71 78.46 10.25 11.56 K = 15

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issue for robotics assembly.The robotics travel routing is based on the TSP method; however, the TSP method focuses on the ® xed location solution and only considers the movement of robotics, but not the movement of boards and magazines. During the robotics assembly, the location of insertion points and magazine slots changes all the time according to the simultaneous move-ment of robotics, board and magazine.To aim directly at the problem of robotics travel routing with changing coordinates, a SA-based approach has been presented in this study to solve the problem.The simulation results demonstrate that theSAapproach is more e cient than

Wang’s approach in all tested cases.Also, the more inser-tion points and/or number of parts, the better the per-formance.

TheSA algorithm is a stochastic heuristic approach to avoiding the trap in local optimal. This study demon-strates that theSAapproach is more suitable for solving the problem of occasionally changed coordinates than other existing approaches. The proposed approach can also be applied to theFPPmodel.Although the proposed approach takes some time program ( each data set takes about 5min to run in a pentium-133PC) , however, the saving of cycle time using the SA algorithm is still a signi® cant improvement.

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