Flow Path Design for an Automated Guided Vehicle System Diane Riopel Andre Langevin Gilles Savard
GERAD and Departement de Mathematiques et de Genie Industriel Ecole Polytechnique de Montreal C.P. 6079, Succursale \Centre-Ville" Montreal (Quebec), H3C 3A7, Canada
July, 1998
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Abstract
An FMS environment requires a exible and adaptable material handling system. Automated guided vehicles (AGV) provide such a system. One of the components of the network design consists of locating the Pick-up and Delivery (P/D) stations on the guidepath network. In this paper we present integer linear programming formulations to solve optimally the P/D station location problem and we discuss the quality of the bounds generated by the linear relaxations of the formulations. The models presented herein are easy to implement and allow solving problems of realistic size in manufacturing.
Resume Un environnement d'atelier exible requiert un systeme de manutention exible et adaptable. Les chariots automatiques sont une solution adequate. Une des dernieres etapes du design d'un systeme consiste a localiser les postes d'entree/sortie sur le reseau de guidage. Cet article presente des formulations de programmation lineaire en nombres entiers pour resoudre de fa con exacte le probleme de localisation des postes d'entree/sortie et nous discutons de la qualite des bornes generees par les relaxations lineaires des formulations. Les modeles sont faciles a implanter et permettent de resoudre des problemes de taille industrielle.
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1 Introduction In the factories of the future, Automated Guided Vehicle (AGV) systems are expected to be in widespread use as material handling equipments. These vehicles transport tools and materials between dierent workcells in exible manufacturing systems. AGVs are programmable and each vehicle is independently addressable. These characteristics confer exibility and adaptability to the material handling systems. The AGVs circulate on a network of guidepaths connecting the various workcells. The unit loads transported by the AGV are picked up or delivered to the workcells at load transfer points (also called Pick-up and Delivery (P/D) stations) located on the path network. In the design of AGV systems, ve types of decision problems can be identi ed: network layout design, load transfer point locating, eet sizing, unit load sizing, and vehicle management. For recent reviews on AGV research, see Co and Tanchoco 2] and Ganesharajah et al. 5]. In 2] a summary of the works on vehicle management (dispatching, routing, and scheduling of vehicles) is presented. The intention of 5] is to unify all the Operations Research issues related to AGV systems and to present a holistic view of the interrelations between them. This paper addresses the problem of locating the load transfer points to minimize the total traveled distances. The location of P/D stations inuence signi cantly the trac intensity and the distances between the workcells. The P/D station locations have a direct impact on the system operating costs. The initial assumptions are the following. The guidepath network is given and allows transportation between any pairs of P/D stations. The orientation of each segment (assuming a unidirectional guidepath) has been determined. The frequencies of ow between the workcells are known. The frequencies include the empty vehicle ow. The quantity of empty ow is dicult to evaluate because the handling requirements in real time are unknown at this step of design. The empty movement requirements can be obtained through simulation using given dispatching rules or can be estimated using the methods of Maxwell and Muckstadt 9] or Egbelu 4]. Similarly, the questions of congestion and blockage in the system are usually handled at the operational level (often in real time). Those questions cannot properly be addressed without knowing the vehicle dispatching rules. Very few papers have proposed exact approaches for solving the problem of locating the transfer stations. Gaskins and Tanchoco 6] nd the ow path (and the related directions) that minimizes total travel of loaded vehicles given the location of the P/D stations. Rabeneck et al. 11] building on Gaskins and Tanchoco work incorporate the locating of the P/D stations via a heuristic. Goetz and Egbelu 7] present an integer linear programming formulation for the guidepath layout and P/D location problem. However their model allows solving only very small problems. Sinriech and Tanchoco 13] consider incorporating intradepartmental ows in formulating the problem of determining the P/D stations in the design of single-loop AGV system. Kim and Klein 8] develop two heuristics for the determination of P/D locations for an AGV system.
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Riopel and Langevin 12] propose a model to optimize the location of material transfer stations within layout analysis. The aim of their approach was to be able to evaluate dierent facility layouts based on the actual distances using the layout aisles. The authors suggested that their method could be used in the AGV system context without detailing the procedure. The purpose of this paper is to extend the model in 12] to that context. The model has also been modi ed to improve its eciency. In Section 2, we present three integer linear programming formulations and we discuss the quality of the bounds obtained from the continuous relaxations. In Sub-section 2.1, we show that when the potential locations for the P/D stations are de ned by continuous intervals along the guidepath segments, it is sucient to consider only a discrete set of these potential locations, thus enabling the use of our formulations in that case also. Section 3 presents numerical experimentations and a conclusion follows.
2 Mathematical Models In this section, we present three dierent linear mixed integer formulations for solving the P/D station location problem. They dier by their sizes and the quality of their linear relaxation bounds. Let de ne the following sets: ( ) : set of feasible pick-up and delivery locations for workcell : distance between locations and = matrix of exchanged ows between workcells ( denotes the ow from workcell to workcell and ( ) 2 indicates that there exists a positive ow from workcell to workcell ) and the following decision variables: 8 1 if location is chosen as pick-up station, < 2 () 8 =: 0 otherwise 8 1 if location is chosen as delivery station, < 2 ( ) 8 =: 01 otherwise ow is carried between locations and = 0 ifotherwise C l
l
dij
i
j
F
fkl
l
k l
k
F
k
l
i
yi
i
zj
j
C l
l
j
C k
k
i
xij
j
With this notation, we proposed the following 0-1 linear programming model (MIP1) to minimize the total distance of the material movement. This model is a re nement of the one proposed in Dahal 3]. (MIP1) X X min (k l)
2F (i j )2C (k)C (l)
fij dij xij
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X
s.t.
2C (k)
i
X
j
X
2C (l)
2C (k)C (l)
X
(i j )
j
2C (l)
X
2C (k )
yi
=1 8
zj
=1 8
3 (1)
k
(2)
l
xij
= 1 8( )j
xij
=
Sk yi
8i of the workcell kjf
xij
=
El zj
8j of the workcell ljf
i
yi z j xij
k l
2 f0 1g 0 8i
fkl >
8i
0
(3) kl
>
0
(4)
kl
>
0
(5) (6) (7)
j
j
where is the number of non-zero terms of line of the matrix and is the number of non-zero terms of column of the matrix . Constraints (1) and (2) ensure that only one pick-up and one delivery locations are chosen for each workcell. Constraints (3) impose that the ow between two workcells is completely sent. Constraints (4) guarantee that no ow can originate from a pick-up location that is not chosen and, if the pick-up location is chosen, all the ow from the corresponding workcell toward all the others should originate from that location. Constraints (5) are equivalent to constraints (4) but for the delivery stations. Note that if the pick-up and delivery variables are xed to a binary value, the resulting problem is a transportation one with capacity constraints on the arcs therefore, the integrality conditions on the ow variables are redundant (and generally deteriorate greatly the time required for solving the corresponding program). The objective function looks only at the variable transportation costs. However, it could easily be modi ed to account for the pick-up and delivery station set-upPcosts. If the set-up cost of each pick-up or delivery station is independent, the terms + 0 , where and 0 are respectively the pick-up and the 2 () delivery station set-up costs, can be added to the objective function. In the case where the set-up cost is dierent whether the station is selectedPas a pick-up, a delivery, or a pick-up 00 and delivery station another term could be added: where 00 is the dierential Sk
k
l
F
El
F
x
ci yi
i
c i zi
ci
ci
C l
i
2C (l)
c yi zi i
c
i
cost between + 0 and the combined station cost. This quadratic term can be linearized in the usual manner. In all cases, the method presented hereafter remains valid. This program can be solved using any branch-and-bound algorithm. It is well known that one factor that aects the eciency of a branch-and-bound algorithm is the quality of the relaxation bound. For this formulation, the lower bound obtained by the linear relaxation is generally weak and the size of problem that can be eciently solved in practice is therefore limited. An equivalent but tighter mixed-integer formulation (MIP2) can be obtained by replacing the aggregate constraints (4) and (5) with the following ones: ci
ci
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xij xij
i
j
kl
kl
4
0 0
(8) (9)
>
>
Again, this program can be solved by a branch-and-bound algorithm with a better relaxation bound but at the cost of solving much larger relaxation subproblems. Although (4-5) and (8-9) give the same set of feasible solutions, (8-9) give in + a much smaller feasible set than (4-5) and hence the ability to solve the formulation with (8-9) is remarkably better than with the more compact formulation that uses (4-5) (see Nemhauser and Wolsey 10]). The number of aggregate constraints (4) and (5) is of the order of the number of possible transfer stations while the number of constraints (8) and (9) is of the order of the square of this last number. Hence, balance between quality of the relaxation bound and the time of resolution of the enlarged problem should be studied. Following techniques used in location theory, it is easy to see that constraints (8) and (9) are valid strong cuts for MIP1. Moreover, in practice, very few of these strong inequalities will be active at the optimum. It is sucient to generate these cuts only as needed. So we de ne MIP3 as being MIP1 plus the inequalities of (8) and (9) that are violated by the solution of the linear relaxation of MIP1 (the cuts are added iteratively to the relaxation until no more are violated). The following cutting-plane/branch-and-bound algorithm generates in Phase 1 these cuts, and in Phase 2, solves the resulting program by branch-and-bound. Phase 1: 0) De ne MIP1 as MIP3. i) Solve the linear relaxation of MIP3. ii) If some inequalities of (8) are violated by the current relaxation solution, add it to MIP3 and go to i otherwise, go to Phase 2. Phase 2: Solve MIP3 by a branch-and-bound algorithm. The following proposition gives the relations between the relaxations of the three MIPs. R
Proposition 2.1
and MIP3, then
If z 1
lp
z
2
lp
andz
n
3 are the linear relaxation values of MIP1, MIP2, lp
z
3 2 1 lp
z
lp
z
lp
2.1 The Continuous Case
In the preceding formulations, all the potential sites were de ned by the set of location ( ) for each workcell . In some instances, the potential locations could be anywhere along several continuous intervals instead of among a number of discrete location. Figure 1 illustrates such a setting. It can be proven though that it is sucient to consider only the locations at the end of the intervals or in a position facing the end of another interval, thus enabling the use of the previous formulation in all cases. C l
l
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30
5
29
28
C6
C5
33 27
25
26
24
22
23
37
C4 20
C7
19
34
21 17
18 15
16
14 36
C3
35
13
5
8
9
4
C2
C1 10
11
12
7
6
1
3
2
Figure 1: Problem A Layout Proposition 2.2 When the intervals of admissible locations are disjoint and each interval is situated along a single edge of the network, a set of optimal locations is composed of points that are at the ends of the intervals. Proof
Consider a P/D station to be located inside an interval with all other points xed. Let ; et + be the two ends of the interval and for any point in the interval let be its distance to ; . Let be the ow between the workcell to be located on the interval and the workcell , and ( ) be the shortest path distance between and the P/D station of workcell located in on the network. We have I
I
x
I
fi
i
di x
i
x
Pi
( ) = minf
di x
where
d
d
;+x i
+
d
i
+ ; g for 0 L
x
; and d+ are the length of the shortest path between P i
i
x
i
and
I
L
(10)
; and I + respectively.
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The function ( ) is continuous, concave, and positive on 0,L]. And so is Therefore the optimum location is at one end of the interval. di x
6
P i
fi d i
.
When the intervals are not disjoint on some edges of the network, a set of optimal locations is composed of points that are at the ends of the intervals or inside the intervals in a position facing the end of another interval.
Proposition 2.3
This proposition can be proven using the same reasoning as in the proof of Proposition 2.2.
3 Numerical Examples
The three formulations have been tested on two problems. The rst Problem (A) is taken from 1], the second one (B) from 3]. Figures 1 and 2 present the respective layout. Tables 1 and 2 give the frequencies of ow between the workcells. Tables 3 and 4 give the length of the segments. The numerical results are presented in Table 5 for Problems A and B. The optimal values of the three relaxations are given in column and the integrality gap in column Gap. The numbers of nodes, pivots and the CPU time (in seconds) are given in the next three columns. The total number of cuts in MIP3 is given with the number of times the linear relaxation of Phase 1 is solved. zilp
Table 1:
Frequencies of ow for Problem A
From-to 1 2 3 4 5 6 7 1 - 4 1 2 2 - 4 5 3 - 8 7 4 - 4 6 5 6 6 5 7 6 5 The problems were solved on a PC-pentium 150 MHz with the optimizer CPLEX 4.0. We used a depth- rst approach for the selection strategy and the pseudo-cost criterion for the branching one, which allow solving problems of realistic size. The important gaps observed are an indication of the diculty of the problems. These results show that the cutting-plane/branch-and-bound approach outperforms the others. For the A Problem, 51 cuts have been generated to obtain MIP3 in ve iterations on a possibility of 699, and 107 over 2280 for the B Problem in seven iterations. Table 6 presents the selected (optimal) P/D stations for the two problems.
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Table 2:
From-to 1 2 3 4 5 6 7 8 9 10 11 12
7
Frequencies of ow for Problem B
1 2 3 4 5 6 7 8 9 10 11 12 30 20 45 20 30 30 20 5 10 - 60 25 45 - 10 20 - 50 20 35 5 20 5 15 10 10 10 40 -
10
11 17
9
12
C2
C1
1
6
19
13
24
20
8
25
18
28
C3
2
23 14 21 22 26 77 78 69 75 80 79 76 81 32 5 7
4
3 65
68
67
C10 61 60
56 64
31
82 33 34 46
51
39 40
45
59 50 44 58
37 36 43
C6
C7 49
35
38
48
C8
30
C5
C12
52
29
C4
15
70 73 74 47 54 62 63 55 53 66
C9
16
C11
71
72
27
57
Figure 2: Problem B Layout
41
42
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Length of the segments for Problem A ( ) ( ) ( ) ( ) ( ) ( ) 18 (37,33) 49 (29,28) 30 (30,34) 24 (18,17) 5 (28,22) 97 (20,19) 12 (17,14) 20 (37,27) 85 (19,18) 22 (14,13) 8 (37,20) 121 (7,6) 20 (13,4) 33 (37,34) 121 (8,5) 70 (13,36) 132 (34,35) 18 (16,19) 14 (27,26) 23 (35,36) 71 (18,15) 30 (26,25) 11 (33,32) 32 (27,24) 15 (25,22) 15 (32,31) 26 (26,23) 76 (22,21) 8 (31,30) 27 43 (21,14) 34 (30,27) 83 32 (30,29) 42 (30,20) 119
Table 3:
( ) (4,3) (3,2) (2,1) (1,6) (6,5) (1,12) (12,11) (11,10) (10,9) (5,36) (9,36) (36,37) i j
l i j
i j
l i j
i j
l i j
i j
Length of the segments for Problem B ( ) ( ) ( ) ( ) ( ) ( ) 10 (61,62) 15 (40,41) 20 (55,63) 10 (64,57) 53 (41,42) 35 (71,61) 21 (57,58) 20 (62,70) 15 (42,43) 21 (58,59) 15 (62,54) 14 (43,36) 5 (59,60) 40 (63,66) 10 (50,49) 5 (60,72) 18 (66,64) 10 (49,48) 10 (44,45) 30 (64,50) 82 (48,53) 5 (45,40) 22 (53,47) 18 (47,46) 7 (45,39) 15 (53,74) 16 (46,33) 6 (39,38) 10 (53,78) 39 (46,34) 14 (38,34) 15 (78,79) 10 (11,10) 47 (38,33) 10 (74,73) 10 (10,1) 17 (33,32) 10 (73,70) 11 (3,67) 5 (32,26) 8 (73,54) 10 (4,68) 10 (26,27) 10 (54,55) 5 (6,8) 10 (34,35) 20 (70,69) 10 (5,7) 21 (35,36) 35 (69,75) 19 (16,76) 10 (36,37) 6 (69,5) 19 (15,77) 20 (37,30) 14 (75,76) 5 (12,18) 98 (30,31) 42 (13,19) 0 (23,27) 28 (31,26) 19 (21,79) 0 (22,80) 43 (37,29) 35 (32,81) 0 (33,82) 10 (29,28) 40 (48,52) 0 (51,49) 19 (28,24) 15 (66,56) 0
( ) 119 6 27 0 0 0 0 0 0
l i j
Table 4:
( ) (1,2) (3,4) (4,5) (4,75) (16,15) (15,14) (14,78) (21,22) (22,26) (23,24) (24,25) (25,11) (25,17) (17,18) (18,19) (19,20) (20,21) (5,6) (6,9) (9,1) (2,65) (2,72) (65,67) (72,61) i j
l i j
i j
l i j
i j
l i j
i j
( ) 0 0 10 10 20 20 10 10 15 20 20 91 0 0 0 0 0 0 0 0 0 0 0
l i j
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Numerical results Problem Formulation GAP(%) Nodes Pivots CPU # Cuts # Phase 1 A MIP1 7451.0 43.46 472 385 2.00 = MIP2 8536.1 25.22 1298 66 2.72 10689 MIP3 8641.1 23.70 185 50 0.72 51 5 B MIP1 61985.0 40.19 20671 4956 55.82 = MIP2 66770.0 30.14 26974 1393 175.59 86900 MIP3 67330.0 29.07 5155 993 18.08 107 7 Table 5:
zilp
Z
int
Z
int
Table 6:
Optimal P/D stations
Problem Workcell # Pick-up station Delivery station A 1 7 2 5 6 3 13 16 4 24 24 5 27 27 6 30 33 7 37 36 B 1 6 6 2 8 7 3 22 4 27 26 5 36 6 38 39 7 45 50 8 53 54 9 63 10 69 65 11 73 75 12 80 80
4 Conclusion In the area of AGV system design, many planning problems can be modeled as integer linear programs (ILPs). However, up to now very few exact methods have been proposed to solve those ILPs. The works found in the literature present only heuristic methods. In this paper, we have presented an ecient exact method based on a cutting-plane/branchand-bound method that enables solving ILP formulations of realistic size in industrial settings for locating the P/D stations. An area worth investigating with this type of methods is the simultaneous determination of both the orientation of the path segments and the locations of the P/D stations.
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Acknowledgments: This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References 1] Bozer, Y.A. and Srinivasan, M.M., Tandem Con gurations for Automated Guided Vehicle Systems and the Analysis of Single Vehicle Loops, IIE Transactions, Vol. 23, pp. 72{82, 1991. 2] Co, C.G.and Tanchoco, J.M.A., A review of research on AGVS vehicle management, Eng. Costs Prod. Eco., Vol. 21, pp. 35{42, 1991. 3] Dahal, A., Design de circuits de chariots automatiques dans un atelier exible, M.Sc. thesis, E cole Polytechnique de Montr eal, 1996. 4] Egbelu, P.J., The use of non-simulation approaches in estimating vehicle requirements in an automated guided vehicle based transport system, Material Flow, Vol. 4, No. 1 and 2, pp. 53{64, 1987. 5] Ganesharajah, T., Hall, N.G. and Sriskandarajah, C., Design and Operational Issues in AGV-Served Manufacturing Systems, to appear in Annals of OR | Mathematics of Industrial Systems, 1996. 6] Gaskins, R.J., Tanchoco, J.M.A., Flow path design for automated guided vehicle systems, Int. J. Prod. Res., Vol. 25, No. 5, pp. 667{676, 1987. 7] Goetz, W.G. and Egbelu, P.J., Guide path design and location of load pick-up/dropout points for an automated guided vehicle system, Int. J. Prod. Res., Vol. 28, No. 5, pp. 927{941, 1990. 8] Kim, J. and Klein, C.M., Location of departmental pickup and delivery points for an AGV system, Int. J. Prod. Res., Vol. 34, No. 2, pp. 407{420, 1996. 9] Maxwell, W.L. and Muskstadt, J.A., Design of automated guided vehicle systems, IIE Transactions, Vol. 14, No. 2, pp. 114{124, 1982. 10] Nemhauser, G.L. and Wolsey, L.A., Integer and Combinatorial optimization, John Wiley and Sons, New York, New York, 1988. 11] Rabeneck C.W., Usher, J.S. and Evans G.W., An analytical model for AGVS design, International Industrial Engineering Conference and Societies' Manufacturing and Productivity Symposium Proceedings, Toronto, May, pp. 191{195, 1989. 12] Riopel, D. and Langevin, A., Optimizing the location of material transfer stations within layout analysis, Int. J. Prod. Eco., Vol. 22, pp. 169{176, 1991. 13] Sinriech, D. and Tanchoco, J.M.A., The centroid projection method for locating pickup and delivery stations in single-loop AGVS, Journal of Manufacturing Systems, Vol. 11, No. 4, pp. 297{307, 1992.
