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Unformatted text preview: :NT. J. rnoo, Res” 1987. Von. 25.'\'o. 5. HGT—mo Flow path design for automated guided vehicle systems R. J. GASKINST and J. M. A. TANCHOCOi‘ An important. factor in the design of automated guided vehicle systems
(AGVS) is the ﬂow path design. This paper presents an approach to determin
ing the optimal ﬂow path. The objective is to ﬁnd the flow path which will
minimize total travel of loaded vehicles. The problem is formulated as a zero—
one integer program. Examples are presented to demonstrate the approach. 1. Introduction Automated guided vehicles are unmanned vehicles used to transport unit
loads, large or small. from one location on the factory ﬂoor to another. These
vehicles are operated with or without wire guidance and are controlled by a com
puter. A system controller is responsible for the regulation of trafﬁc when more
than one vehicle is in the system. In addition to vehicles and controls, a third major component of an AGVS is
the flow path, The direction of traffic along the flow path can be deﬁned as either
unidirectional or bidirectional. Unidirectional ﬂow results when vehicle travel is
restricted to only one direction along a given segment of the ﬂow path. Bidirec
tional flow. on the other hand, occurs when ﬂow exists in two directions. Bidirec
tional flows have the advantage of reduced travel time. ‘Nith unidirectional
travel, a vehicle may have to travel a greater distance in moving from one point
to another than it would if bidirectional ﬂow is allowed. On the other hand, uni—
directional ﬂows require fewer controls and are more economical. When designing an automated guided vehicle transport system, there are
several important issues to consider. Along with the choice of unidirectional or
bidirectional ﬂows= the question of overall ﬂow path design will have to be
addressed. Also, the type and number of vehicles to be used have to be deter—
mined. The scheduling rules to apply need to be studied in detail. This paper is concerned with the issue of ﬂow path design. The procedure
presented in this paper is a zeroone integer programming approach to developing
an optimal ﬂow path design. Only the case of unidirectional ﬂow paths is con—
sidered. The strengths and weaknesses of the procedure are discussed. 2. Review An automated guided vehicle system, like all other material handling systems,
is concerned with the movement of material. In order for the vehicle system to be
effective. the ﬂow of material needs to be as efﬁcient as possible. In past studies,
various approaches have been taken to meet this objective. Maxwell and Muck
stadt {1982) developed a method of simultaneously determining the minimal Received June 1986.
T School of Industrial Engineering. Purdue University. “lest Lafayette, lndiana 47907.
USA. 668 R. J. Gaskins and J. 11!. A. Tanshoco number of vehicles required and the vehicle routes so that the required material
handling load is satisﬁed. Blair et a2. {1985) also addressed the problem of vehicle
routing. In their study, integer programming techniques were used to determine
the routes of the vehicles. The Objective of their approach was to develop a
vehicle routing schedule resulting in the maximum distance that a vehicle travels
being minimized. In both of the above two studies, an effort was made to reduce vehicle travel.
Minimizing vehicle travel is also the objective of the approach presented in this
paper. However, there are some important differences. First, in this approach
only the movement of loaded vehicles is considered. No effort is made to reduce
the travel of unloaded vehicles. A second major difference in this approach is the
assumption that the ﬂow path design does not exist. In fact, the goal of the
procedure is to produce the optimal ﬂow path. In the papers discussed above, it
was assumed that the ﬂow path was given. The procedure presented in this paper assumes that vehicle travel is restricted
to certain areas. To travel from one point in the layout to another, it is clear that
the shortest distance would be a straight line from the ﬁrst point to the second.
This straight line distance is known as the euclidean distance (Francis and W'hite
1974). However, it may be impossible for the vehicle to travel this path. In a
departmental layout, for example, vehicle travel is conﬁned to the aisles. A second measure of distance is rectilinear distance (Francis and “White 1974).
For a two dimensional layout, the rectilinear distance between two points is
deﬁned as the sum of the absolute difference in at coordinates of the points and
the absolute difference in the y coordinates of the points. Given rectangular
departments, this is an appropriate measure. However, when trying to minimize
vehicle travel in a unidirectional environment, rectilinear distance is not always
acceptable. In the unidirectional case, a vehicle may have to travel further than
the rectilinear distance to get from one point to another. As neither euclidean distance nor rectilinear distance is applicable in every
case, an effort needs to be made to minimize path distance. Path distance can be
deﬁned as the distance a vehicle travels along a feasible path while moving from
one point to another. The path may be straight line, rectilinear, or of some other
form depending upon the location of the points and the shape of the departments. 3. Flow path determination The objective of the mathematical programming procedure is to ﬁnd the uni—
directional ﬂow path which, when the shortest routes are taken, will result in
total loaded travel being minimized. The approach taken involves zero—one
integer programming. rI‘he formulation of the program and an example problem
are presented. Before the formulation can be developed, the following information has to be
obtained: a layout of departments, aisles, and pick—up (P) and delivery (D) sta—
tions, location of these P/D stations, and a from—to chart containing the material
ﬂow intensities between departments. The next step is to translate the above
information into a node—arc network. Nodes are used to represent corners, inter
sections and P/D stations. while arcs represent possible directions of travel along
the aisles. In order to identify them, each node is assigned a number. An arc is
identiﬁed by a variable 33,}. For example, the are representing travel from node 1 Flow path design for AG V8 (in!) 10 (a)
Figure l. (a) Example layout. {b} Corresponding network for example layout. (c) Flow
path for example layout. to node 2 is assigned the variable 9:12. Furthermore, these variables are zero—one
variables. If a?” = 1, then this arc is included in the ﬁnal flow path design. If
ailJ : 0, the arc is not included. 3.]. Simpliﬁed example Consider the simple departmental layout shown in Fig. l (a). The correspond—
ing node—arc network is shown in Fig. 1 (b). In Fig. 1 (b), 01,} represents the dis
tance between adjacent nodes. P1 is the location of pick—up station 1 and D2 is
that of delivery station 2. The ﬂow intensity for this problem is 100. The formulation of the objective function is based on the fact that to travel
from P1 to D2, the guided vehicle must exit. on x32 or x36 and enter D2 on x45 or
2:65 . Because unidirectionalittr is assumed, there are only four distinct pos
sibilities travel on x32 and x45 , or an and $55 , or 3:35 and x45 , or 2:35 and x65 . The next step is to consider each of the combinations separater and ﬁnd the
corresponding shortest path from P1 to D2. For example, if the guided vehicle
leaves P1 on x32 and arrives at D2 on x45 , then the shortest possible path is node
3 to node 2 to node 1 to node 4 to node 5. This path has a distance of 24. For the
3:32—2:65 , 3735—145 , and $36—$65 combinations, the corresponding shortest path dis
tances are 52, 36 and 10, respectively. lWith these distances, along with the ﬂow intensity, the objective function can
be stated as follows Minimize 100[24(x32x45} + 52(x32 x55} + 36(93352745) + 10(3236 3:65)] Given that the :1: variables are zeroone variables and that only one of the four
combinations can be chosen, only the chosen combination will have a product of
1. All other combinations will have a product of zero. Thus, in order to minimize
the objective function, the combination with the shortest distance wiil be select.—
ed. The key to the above procedure is that one and only one of the four com—
binations has a product. of 1. To ensure this, constraints need to be added to the
formulation. To guarantee unidirectional travel, a constraint needs to be added for each
pair of adjacent nodes. For the above problem. the following are needed: (i?!) ' R. J. Gaskins and J. M. A. Tanchoco 3314+x41 ='
$47 +xv4=l 3518 +3787 =' By setting the right—hand side equal to 1, one and only one are. will be chosen
from each pair. Also required is that each node is reachable. However, nodes cannot become
sink nodes. In other words, each node must have at least one incoming arc and at
least one outgoing are. Considering node 4, the following constraints ensure that
the ahoveis true: 3741+3545 ‘l‘ 3’47 21
I'14'l'53:544'5171'4?1 Similar constraints need to be added for each node in the network. Finally, constraints need to be added to prevent the situation where a group
of nodes become a sink. Also, the same group of nodes needs to be reachable. In
the given network, if :74 and I“ are in the ﬁnal design, then nodes 1, 2, .. ., 6
would act as a sink since once a guided vehicle enters this segment of the network,
it can never leave. 0n the other hand, if 1:“ and x58 were both in the ﬁnal design,
nodes 1, 2, 6 could never be reached. To keep these two problems from
occurring, the following two constraints are included: 3774 '9' x86 21
5541 +3753 3' The three types of constraints discussed thus far are basically all that are
needed to produce the desired results. In more complex layouts, extra constraints
are needed to ensure that the shortest routes are always taken. Also, extra con
straints are needed to rem0ve the non—linear terms in the objective function.
Translating the problem from a nonlinear to a linear one simpliﬁes the solution
procedure. The solution to the example problem is 332:1 3541:] 3723:]
3563:] 336:1 337:1
xes=1 “774:1 354:1 all other variables = 0 The problem was solved using the Multipurpose Optimization System computer
package (Cohen and Stein 1978). A direct search algorithm is used in this
program, The above solution yields the optimal ﬂow path design. This optimal
ﬂow path design is shown in Fig. 1 (c). If the guided vehicle uses this ﬂow path
and always takes the shortest route from P1 to D2, total loaded travel will be
minimized. Given that the vehicle travels from P1 to D2 on x36 and 3365, it can be seen
that loaded vehicles will never travel on arcs .1363. x87, and 1'74. Therefore, these
arcs could be eliminated from the ﬂow path design. This would result in a Flow path design for AGVS [i7] reduction of ecsts in terms of vehicle control costs. On the other hand, it may be
desirable to include these ares as an alternate route for unassigned empty vehicles
when 1365 is blocked. If it is decided that. the unused arcs are not necessary, then two approaches
can be taken to remove them. The ﬁrst approach involves changing the formula—
tion of the problem. By changing the unidirectionality constraints from strict
equalities to less than or equal to constraints, the need for every pair of adjacent
nodes to join is eliminated. However, in some cases, including the example
problem, this method fails. In the example, every node is included in at least one
of the shortest paths used in constructing the objective function. Given this fact
along with the constraints which ensure every node is reachable, the new formula—
tion will produce the same ﬂow path design. If it were the case that the nodes were not included in some shortest path,
then the constraints ensuring that these nodes are reachable can be removed.
Unfortunately, this results in alternate optimal solutions. The second approach to eliminating unused arcs is to solve the original formu
lation and produce the flow path design. Then, by examining the from—to ﬂow
intensity chart, it can be determined which arcs will not be used by loaded vehi—
cles. These arcs are then removed from the ﬂow path design. 3.2. More complex example Although the previous example was useful for developing the formulation, it is
not typical of a real world departmental layout. The case just presented consisted
of only two departments and flow existed only between two stations. Normally,
an automated guided vehicle would be used in an environment made up of several
departments and ﬂow would exist between several sets of stations. _ The procedure described above can be easily extended to these more typical
and more complex layouts. The major difference is in the formulation of the
objective function. For each entry, an expression similar to the objective function
in the simple example is developed. The sum of these expressions is the objective
function. The procedure for developing constraints remains the same with the
addition of the extra constraints required to ensure that the shortest route is
always taken. "l‘o demonstrate the application procedure to a larger problem, a formuiation
was developed for the node—arc network shown in Fig. 2. This layout consists of
six departments, a receiving station, and a station for sending material out of the
facility. Flow exists between 11 pairs of stations. The entire network consists of
22 nodes. The additional constraints mentioned above are required for this example.
Before these constraints can be presented, though, more insight into how the
formulation works is needed. Xotice ﬁrst that only a few of the many variables
appear in the objective function. Since the remaining variables all have coefﬁ—
cients of zero, their ﬁnal values [i.e. whether or not they are in the final ﬂow
path] have no effect on the value of the objective function. With this in mind, it
seems as though the only arcs which are inciuded are those which are actually in
the objective function. All other arcs, then, can be included or not included so
long as the constraints are satisﬁed. (572 R. J. Gcskins and J. ile. A. Tanciwco Figure 2. Network of 22—nodc problem. For each pair of arcs in the objective function, there is a corresponding short
est path. When a path (i.e. pair) is selected, those arcs which do not appear in the
objective function must be included in such a way that the shortest path is
included in the guide path. In the ﬁrst example. this condition is guaranteed. Due
to the effect. of the constraints, selecting one of the four pairs of variables also
determines the value of all other arcs along the shortest path and the desired path
is included in the guide path. This is not the case in the larger example. Here, there are variables whose
values are not determined as a result of determining the values of the variables
which appear in the objective function. These undetermined arcs present a
problem if they are part of a shortest path. In travelling from node 2 to node 9
using ares x21 and 3:39, the shortest path is .1721, 3:16, arm, as”, and 2:39. In this
path, In, 3:39, and 1316 are determined while at“ and .1775 are undetermined.
Therefore, an extra constraint is needed to ensure that 276; and 2373 are included if
this path is selected. This constraint is as follows 3321+ 35539 — 3716 — x57 — 3773 g 3 In general, for each shortest path found in constructing the objective function
the following constraint. is added: Flow path design for A0 VS “73 R + 9 SHIPPING Figure 3. Optimal ﬂow path f01'22node problem. (it  2)(ﬁrst are} + {a — 2)(last are) — (sum of the intermediate ares) 5H (n — 2)
where n is the total number of arcs in the path.
rI‘he above constraint is required whenever there is an undetermined are in a
shortest path, In those paths where there are no undetermined ares, this con
straint is optional. Finally, there may be instances where there are undetermined
ares which do not. appear in any of the shortest paths. No constraints are needed
for these ares, However. the existence of such ares results in alternate optimal solutions.
The optimal ﬂow path for the 22—node problem discussed above is shown in Fig. 3. 4. General formuiation The general formulation of the ﬂow path design problem can now be stated
based on the following deﬁnitions of the decision variables and parameter values. 4.1 . Variables { 1 If the are from node 12 to nodej x“. —_— . is included in the layout, 1 0 otherwise (in R. J. Gaskéns and .1. M. A. Tamhose 4.2. Parameters dmp distance from nodem to node as using path 30
f,“ ﬂow intensity from node m to node 3*: RP total number of arcs in path p S set of nodes such that each node in S is adjacent to some other node in S 4.3. Objectieefunction 4
Minimize Z 2 fm Z [dmp at," arm] q, r E p; Vf
m n l 4.4. Constraints
(a) Unidirectionality constraints
xu+xﬁ=l W,j
(b) At least one input are 2%;1 Vj Vi
adjacent to i (6} At least one output are E 3:” ? 1 Vi
VJ“ adjacent toj
(:1) At least one input are to a group of nodes Exp} 2 l a" = nodes adjacent toj Vj e 8; VS
(e) At least one output are to a group of nodes 2 xi}. ; l j“ = nodes adjacent to i
1.. Vi e 8; VS
(f) Ensure shortest path is taken
mp — 2}qu + {up — 2)x,,, — E as“. a n, — 2 V}? Vi. i t m. n
in path p
An optional constraint to force ﬂow balance at fourway nodes is given by Z I”. ; 2 j’ = is a 4way node
j. Vi adjacent to j’ Flow path design for AG VS 67;": 5. Summary The objective of this study was to develop an approach to the design of an
optimal ﬂow path for automated guided vehicle transport systems. A procedure
based on zero—one programming was shown to result in an optimal ﬂow path. The
advantages and disadvantages of this approach are discussed below. The procedure developed was based on mathematical programming tech
niques. An alternative approach would be to use simulation to ﬁnd the best ﬂow
path. This would give a more complete picture of the system. Unlike mathemati
cal programming, simulation would take into account the travel of unloaded vehi
cles, vehicle blocking, and congestion. On the other hand, mathematical
programming gives the solution without having to analyse simulation output.
Moreover, formulating the program takes less time than building a simulation
model and probably could be solved in less time than it takes to make several
simulation runs. The best approach may be to generate a layout using mathe
matical program and then test the design using simulation. Having identiﬁed the strengths and weaknesses of the procedure, the next
step is to consider applications which exploit the advantages. The environment in
which the approach .is best suited is a ﬂexible one, more speciﬁcally, an
environment where it is possible for the ﬂow path to change over time. As the
material flow intensity fromto chart changes, the optimal ﬂow path may change
also. Assuming the departmental layout does not change, the constraints of the
problem remain the same. All that will change is the coefﬁcients of the objective
function. Solving this new formulation would be faster than doing a new simula—
tion study. Acknowledgment
The work reported in this paper was supported in part by 1NSF/Purdue Uni— 1 versity ltngineering Research Center for Intelligent Manufacturing System,
USA, under Grant No. CDR8500022. [7n facteur important dans la conception de systolrnes de véhicules guides
autoinatisés (AGVS: Automated Guided Vehicle Systems} est I’etude du circuit.
d'etoulement. Cet article presents unc stramgie visant s deﬁnir le circuit
optimal aﬁn de determiner le circuit qui permettra de minimiser ie déplacc—
ment total des vehicules charges. Le probléme est formulé comme programme
en hombres entiers zero1m. Des exemples sont fournis a l’appui. Die Projektierung der Fahrbahn stellt einen wichtigen Faktor bei dem
Entwurf automatisch gelenkter F‘ahrzeugsysteme dar. Dieser Beitrag stellt.
cine Metbode zur Ermittlung der optimalen Fahrbahn vor. Die Zielsetzung
besteht. in der Ermittlung der Fahrbahn, die den Gesamtweg aller beladenen
Fahrzeuge minimiert. Das Problem wird als ein ganzzahiiges Xull—Eins— Programm formuliert. Die Methode wird anhand ciniger Beispiele veranschau—
lieht. References BLAIR. E. L,, CHARNSETHIKTIL. P._. and Vasouss. A., 1985, Optimal routing of driverless
vehicles in a ﬂexible material handling system, Texas Tech University and Renssel
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ming (Reading. MA: Addison~VVesley). 1576 Plow path design for AG VS COHEX, 0., and STEIK, J._. 1978, Multi purpose optimization system user’s guide. Manual
No. 320. Vogelback Computing Center, Northwestern University, Evanston, Illinois,
USA. EGBELL‘. 1’. lJ., and TANCHOCO, J. M. A., 1982, Operating considerations for the design of
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