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Unformatted text preview: INT. J. PROD. RES., 1993, VOL. 31, N0. 6, 1387—1407 Economical design of material ﬂow paths KAP HWAN KIMTI and J. M. A. TANCHOCOT“ In this paper, we suggest a branchandbound procedure for designing ﬂow paths
of ﬁxedpath material handling systems. The method may be applied to auto
mated monorail systems (AMS), automated guided vehicle systems (AGVS).
ﬂexible conveyor systems and other ﬁxed~path material handling systems. We
formulate an economic model which considers the construction cost of each path
segment as well as the travel cost. A procedure is developed to determine the
canﬁguration of flow path and the direction or each flow path segment. A tight
lower bound for the optimal objective function and an efficient search strategy
are suggested. Computational performance of this procedure is compared with a procedure previously developed by Kaspi and Tanchoco for the special case of
AGV systems. 1. Introduction When a material handling is carried out automatically on a prespeciﬁed ﬁxed
path, the conﬁguration of the path network and the direction of ﬂow allowed in each
segment of the path network signiﬁcantly affect the operational efﬁciency of the
entire material handling system. For example, in AGVS, the ﬂow path network
affects the total distance vehicles travel to accomplish the material transport tasks.
Furthermore, the conﬁguration of the path network is related to path construction
cost, space cost and cost of controls, etc., which are categorized as ﬁxed costs. A
complicated ﬂow path network increases the ﬁxed cost signiﬁcantly although it may
reduce the operation (travel) cost. Thus, the designer should try to reduce the
operation cost as much as possible while maintaining the ﬂow path conﬁguration as
simple as possible. Provided that a ﬁxed path is used for material handling, the direction of ﬂow
along the path can be deﬁned as either unidirectional or bidirectional. Unidirectional
ﬂow path restricts materials to travel in only one direction along a given segment of
the ﬂow path network, while materials may travel in both directions on bidirectional
path. Since complicated controls are required and blockings between vehicles may
occur frequently, it is considered realistic to utilize bidirectional ﬂow path only in the
aisles where ﬂow intensity is low. In this paper, we consider the two parts of the ﬂow path design problem: (1) to
determine the conﬁguration of the ﬂow path, that is, determine which candidate
segments on the path network should be included in the ﬁnal ﬂow path considering Revision received September 1992.
TSchool of Industrial Engineering, Purdue University, West Lafayette, IN 47907, USA.
IAssociate Professor of Industrial Engineering, Pusan National University, Changjcon dong, Kumjeongku, Pusan 609735, Korea. Visiting Scholar at Purdue University,
1990—1991. *Address correspondence to: J. M. A. Tanchoco, Professor of Industrial Engineering. 0020—7543/93 $1000 © 1993 Taylor & Francis Ltd. 1388 K. H. Kim and J. M. A. Tanchoco the ﬁxed cost and the contribution of each path segment to the transportation cost;
and (2) to determine the direction of each unidirectional path segment. The ﬂow path design problem was ﬁrst formulated by Gaskins and Tanchoco
(1987) for the AGV guide path layout as a zeroone interger programming problem.
In the paper, they considered the unidirectional ﬂow path problem and the decision
variables are the direction of each path segment. Gaskins et al. (1989) extended this
formulation to consider virtualﬂow paths for freeranging AGVS. They formulated
the problem as a mixedinteger programming problem and multiple lanes per aisle
were allowed and ﬂow intensity was restricted by the path capacity. Recently, Kaspi
and Tanchoco (1990) developed a branch—andbound algorithm for Gaskins and
Tanchoco’s problem (1987) which is computationally more efﬁcient. Two papers by Rabeneck et al. (1989) and Goetz and Egbelu (1990) addressed
the problem of selecting guide path as well as the location of pickup and delivery
points of AGVS simultaneously. In both papers, the Gaskins—Tanchoco model was
used as the base model. In the network design research area the main application problems have been
transportation and computer network design, many researches have been done to
optimize the network conﬁguration. And several researchers have proposed various
branch and bound solution procedures for solving network design problems (Dionne
and Florian 1979, Hoang 1973, Los and Lardinois 1982). Magnanti and Wong
(1984) presented a good review paper on network design models and Gallo (1983)
analysed various lower bounds used in branchand—bound procedures. This paper is distinctive in the following aspects compared with previous
researches: o The model for ﬂow path design is generalized by considering the ﬁxed cost
(construction cost, space cost, cost for controls, etc.) as well as the travel cost. 0 Performance of the branchandbound procedure is enhanced by using a tighter
bound and a more efﬁcient search scheme. 2. An economic model for ﬂow path network design The formulation in this paper is based on the model by Kaspi and Tanchoco
(1990). But the ﬁxed cost for constructing each segment of path is considered and
some aisles are allowed to have double lanes of opposite direction. Furthermore,
some aisles may be completely closed in case the traffic rate on the aisles are too low
to pay for the construction cost. Let a network describe the ﬂow path conﬁguration in which the nodes represent
pickup points, delivery points and intersection points, and the arcs represent
candidate segments of path between two adjacent nodes. Each arc is assigned a
length equal to the distance between the nodes it connects. The objective of the formulation is to determine whether to include or not
include a speciﬁc arc in the ﬂow path network and to set the direction of each are so
that the sum of the travel cost of vehicles and the ﬁxed cost for arcs is minimized. The ﬂow requirement between pickup and delivery station is represented in the
form of a from/to matrix which may include both loaded and empty vehicle travels
or only loaded travels. We assume that each ﬂow requirement will be satisﬁed using
the shortest path on the ﬂow path network. Economical design of material ﬂow paths 1389 The following notation is used to describe the model: n = the number of nodes
f,,,, = ﬂow intensity from pickup node I to delivery node m
d” = length of arc (Lj) I if are (if) is included in the path from pickup node I to delivery
){W1 — node m 0 otherwise I if are (if) is included in the ﬂow path network
ZJl = 0 . otherwrse
A = set of all candidate arcs in the original network
A0 = set of arcs for which at most one between arcs (i, j) and (j, i) is allowed
to be included in the ﬂow path network because of space limitations Q = set of pickup/delivery node pairs whose ﬂow values are positive
KU = the ﬁxed cost of are (i, j)
WU = the travel cost per unit load per distance unit on arc (i, j) The objective function is to minimize the sum of travel cost of vehicles and the
ﬁxed cost for arcs, that is Minimize Zflmmm<ZXiﬂmdij>+ZKijZij (1)
LM i,j L] The constraint set includes the following: C1. C2. C3. C4. C5. C6. C7. C8. Ensure the path from pickup node I to delivery node m is feasible
Xiﬂmszlj for all I, m, i,j (2)
Single lane restrictions for some aisles
Zij+Zﬁ<1 for (i,j) and (j, i)eAc (3) At least one input arc 221721 for allj (4) At least one output are 220.21 for all i (5)
j One output are from pickup node I using the path from I to delivery node m ZXmm=l for all I, m (6)
k One input are to delivery node m using the path from pickup node I to node m 2 X km,” = l for all I, m (7)
k Number of input arcs equals to the number of output arcs z XW = Z X W for all j, 1, m (8)
i k Integral constraint 1390 K. H. Kim and J. M. A. Tanchoco Xij,m,Z =Oorl for all 1, m, 131' (9) 1'1
Once Zijs, (i, j)eQ, are ﬁxed, the above problem may be decomposed into
independent shortest path problems each of which corresponds to a (l, m) pair of Q and may be solved easily. We suggest a branch—and—bound technique to enumerate
all the possible values of ZUs. 3. Branch and bound procedure
The overall procedure may be summarized as follows: On a vertex of the search tree, we select an arc for the next branching which has
the maximum difference in the effects of including are (i, j) or excluding are (i, j) in
the ﬂow path network. That is, we foresee one step further the inﬂuence of the next
decision on the total cost. Once we select an arc, we branch the vertex on the search tree and follow a more
promising branch. It is a kind of depthﬁrst search. And then, the lower bound of
the next vertex is calculated. Whenever the lower bound is larger than the current
upper bound, the branch is pruned. When we arrive at the bottom of the search tree, we update the upper bound and
backtrack the search tree. To explain the procedure in more detail, the following additional deﬁnitions are
introduced: Assume we are on a speciﬁc vertex n5 in the search process. A; = set of arcs which are determined to be included
A5 = set of arcs which are determined to be excluded
A? = set of arcs on which no decisions have been made yet.
Note that AzAlsuAquiJ
AS = A'SOA;J
T(As) = the total cost when all the arcs in As are constructed and used
K(As) = the ﬁxed cost when all the arcs in A5 are constructed L(AS,Q)= the travel cost when all the arcs in AS are used to deliver ﬂow
requirements of Q. Note that T(As) = K043) + MAS, Q) (10) a penalty function which represents the increase in the travel cost
resulting from deleting arcs in X from As.
We may redeﬁne it as 7rx(Q)=L(z‘1s—X,Q)—L(AS,Q) (20) (11)
We will use nij (1m) instead of Rim”({(l, m)}) to simplify the notation. ﬂx(Q) 3.1. Selection of are for branching The branching of the search tree on a vertex ns is done using one of the arcs in
AB. Note that the decision to be made is whether an arc in A? should be constructed
or not. First, we evaluate the increase in the travel cost resulting from deletion of arc
(i,j), nil(Q), for each arc (i,j)eA;J. And then, we compare it with the ﬁxed cost, K”, and choose an are for the next
branching which has the largest difference in the two values. The reason we select the
arc with the largest difference is to get a good upper bound as soon as possible. Economical design of material ﬂow paths 1391 3.2. Branching and choosing a branch to follow next During the branching process, whenever we ﬁnd out the arcs which has the only
one choice by constraints (3)—(5), we ﬁx the decision on the arcs to reduce the
number of branches on the search tree. By doing so, we can signiﬁcantly reduce the
number of calculation of the lower bounds which consume most of the calculation
time of the algorithm. Let an are (a, b) be selected in the previous procedure. Then, if Kan>7rab(Q), we
consider the branch of deleting are (a, b) as the more promising one and proceed to
the branch. Otherwise, the branch of constructing are (a, b) will be the next branch
for the evaluation of lower bound. 3.3. Pruning branch At each branching node ns, the lower bound will be evaluated using the
procedure of section 3.5. A branch will be pruned if (1) the lower bound is greater
than the current upper bound or (2) no feasible path is obtained from a pickup node
to a delivery node with positive ﬂow requirement. 3.4. Updating upper bound and backtracking Whenever we arrive at the bottom of the search tree, i.e. there is no arc
undetermined, the objective function is evaluated and compared with the current
upper bound. If it is smaller than the current upper bound, the upper bond will be
replaced by the value of objective function. And then, the backtracking procedure is invoked. The backtracking procedure
returns to the source branch and considers a previously not selected branch of the
source, i.e. a sibling branch, if it is not pruned and has not been chosen before. If the
sibling branch is not available, then backtracking is performed again in the reverse
order of the branching process. The same backtracking procedure will also be employed when a branch is
pruned. 3.5. Lower bound Considering that A, is the set of both included arcs and undetermined arcs, all
the solutions of Zijs below the vertex n, in the searching tree may be represented by
the set of arcs, As—X, where X is a subset of AP. In the following, we will derive a lower bound of T(AS—X) which holds for all X.
From equation (1 1), we get
L(As ~ X: Q): L(Asa Q) + nX(Q) =L(AS,Q)+ Z ”x0119 (i..i)EX >L(AS,Q)+ Z nij(Yij) (12) (Max
where Yijs are a partition of Q, that is, U Y,~,=Q and Yizjz=g’ (i.j)eX for all (iisji) and (i25j2)EX and (i15j1)9é(i2,j2) 1392 K. H. Kim and J. M. A. Tanchoco In Kaspi and Tanchoco’s paper, only L(As, Q) was used as a lower bound for
their branch and bound procedure. Thus, the lower bound of the travel cost in this
paper is tighter than the one of Kaspi and Tanchoco’s by the amount of the second term of (12).
From equation (10) and unequality (12), it follows that T(As—X)=K(As—X)+L(AS+X,Q) >min:K(A's)+ Z K.~j+L(As,Q)+ Z 7:,,.(Y,.,.)] X (LDSAF  X (i,j)eX =K(A;)+L(AS,Q)+min[ Z Kij+ Z nij(1’}j): (13) X (i, DEA? — X (i. DeX Note that the above inequality holds for any partition Yijs.
Let’s assume that YU is given for every (i,j)eA:J and denote the terms inside the brackets of equation (13) as M (X ). Then, M (X ) can be easily minimized by
comparing K” and HUD/U) for each arc (i,j). But for arcs which are restricted to have
at most a single lane in an aisle, Kﬂ+7tij(Y,J) and Kij+1rﬁ(YJ;) should be compared
when both Kij<7tij(Y,j) and K ﬁ<7z ﬁ(Yﬁ) hold simultaneously, and then the smaller
one will be used in evaluating min M(X)
x The ﬁnal problem is how to partition Q into Yijs to make min M(X)
x as large as possible. Let M U be the contribution of are (i, j) to the value of min M(X) X The strategy used in this paper is to give the highest priority on the arc with‘the
largest M ii value and on the ﬂow requirement with the largest contribution to the 7:
value of the arc. The partitioning procedure is summarized in the following: (1) Q’ =Q (2) Calculate M” for all (z',j)eA:J assuming Yij=Q’. (3) Select an arc, (i *, j *), with the next largest M i , value. If the largest M , 1. value
is zero, stop. Otherwise, go to (4). (4) Allocate elements of Q’ to the arc (i*,j*) in the decreasing order of the
penalty cost resultant from the corresponding element. Allocate elements of
Q’ as few as possible as long as the decision on are (i *, j*) to evaluate M i. 1*
remains valid so that the remained ﬂow requirements may be used for other
arcs. The resulting set of allocated elements is YWh Eliminate arc (i *, j *)
from further consideration. Delete the allocated elements from Q’. Go to (2) We will denote the lower bound derived above as V1. Another lower bound
suggested by Los and Lardinois (1982) was Economical design of material ﬂow paths 1393 V2 =maX {IQ/4:), Km} +L(As5 Q) (14) where Km, is the ﬁxed cost of the minimum ﬁxed cost spanning tree of the original
network. Now, we get a new lower bound
V: max (V1, V2)
:ma’x {K(As )+mxin M(X)5 Kmst}+L(A55Q) (15) 3.6. Reachability To guarantee the reachability of empty vehicles from each delivery station to all
the pickup stations, the procedure described in Kapsi and Tanchoco’s paper, which
adds an inﬁnitesimal value to the from/t0 matrix, may be used. As an alternative, we can embed a procedure to check reachability in the
evaluation of the shortest distance matrix which is used to calculate L(AS, Q). In the
course of updating the shortest distance matrix, as soon as we ﬁnd any pair of nodes
between which there is no feasible path, the branch is pruned. 4. Illustration of the algorithm
Suppose that the initial ﬂow path network is given as in Fig. 1(a and c) represents
the ﬂow requirements betwen workstations. For each one solid line in Fig. 1(a), only (a) Initial ﬂow path network /§:\‘\®
®\1/® (b) Shortest path distance matrix Figure 1. Data of the example for illustration of the solution procedure. 1394 K. H. Kim and J. M. A. Tanchoco one lane is allowed for ﬂow path. Note that some aisles have enough space to
accommodate up to two lanes. The ﬁxed cost of a lane and the travel cost are
assumed to be $ (20xlength of the lane) and $1 per unit distance per travel
frequency, respectively. We will illustrate only the ﬁrst iteration of the algorithm. Initially, we have
elements of various sets as follows: A: {(151), (4:1), (1,5),(5,1), (2,4),(412), (2,5), (5,2): (3,5), (5,3), (3,4),(413)} Q={(1,2), (1,3),(2,1), (2,3), (3,1), (3,2)} Selection of arc for branching
First, we compute 7:” (Q) for each (i, j)eAE, and then calculate the difference
between the value and KU. n14(Q)=1t14(12)+n14(13)=(23— 17) x 18+(31 — 19) x 21 = 150 and K14=240. Thus, n14(Q)—K14(=514)= —90. In the same way, we can get 514: —50, 515: —360, 551: —360, 624:68, (342:8,
625: —80, 652: —40, 534: —318,643= —298, 635: —200, 553: —240. Arc (1,5) will be the are for the next branching. Branching and choosing a branch to follow next Since n15(Q)<K15, deleting arc (1,5) is more promising branch to follow next.
Proceed to the branch of deleting arc (1,5). Considering the constraints (375), and the fact that we decided to delete arc
(1,5), arcs (1,4) and (5,1) should be constructed and are (4,1) should be deleted.
Therefore, we ﬁx the decision on arcs (1, 4), (5, l) and (4,1) as above. Pruning branch
We evaluate the lower bound for the current branching node ns as follows: Currently
A;={(1,4), (5,1)}, AE={(1,5), (4,1)}
and
AE= {(2, 4), (4, 2), (2, 5), (5, 2), (3, 4), (4, 3), (3, 5), (5, 3)}
Calculating nij (lm) for all (i,j)EAsU and (l,m)eQ
n25(21)=840 n25(23)=20 113581): 154 n35(32)=60
n42(12)=540 714303): 126 7:52(32)=60 7153(23)=20
From now on, we will evaluate min M(X).
x Let Q’=Q. Then, Economical design of material ﬂow paths 1395 7325(Q’) = n25(2l) + “25(23) = 860 7r35(Ql)=”35(31)+ 7T35(32) =214
7r42(Ql) = 7:4202) = 540 7543(Ql) = 7;4303) = 126
7352(Q')=W52(32)=60 “53(Q')=7‘53(23)=20
Other values of n’s are zero. Since
K24>7t24(Q’) and K42<1t42(Q’) M24=7r24(Q’)=0 and M42=K42= 100
And since
K43 > 7143(Q’) M43 = 7143(Q')=7I43(13)=126
In this way, we get
M35 =7r35(Q’) = n35(31)+n35(32)=214 M25 =K25 = 100,
M25 =7r52(Q')=7t52(32)=60 M53 =7r53(Q’)=7r53(23)=20 Other M z is are all zero.
Thus, (i*,j*)=(3,5), Allocate the elements of Q, (3,1) and (3,2), to are (3,5).
M35 will be ﬁxed as 214. Eliminate arc (3,5) from further consideration. Revise Q’=Q—{(3, l), (3,2)}. Then
7525(Q’)=7125(21)+7525(23)=860 7T42(Ql)=7r42(12)=540
7143(Q’) =7I43(13)= 126 ”53(Q')=7t53(23)=20 Other values of 7: s are zero.
In the same way, we get M43=126 M25=100 M42=100 M53=20
The above procedure is repeated to get all the MUEASU. Finally, we get
M35=214 M43: 126 M25=IOO M42: 100 M53=20 All other values of Mi]. are zero.
Thus minM(X)=ZMij=560 X ij
Considering
L(As, Q) = 2260 K(A;) = 600 Km, = 700
the lower bounds becomes V: {600 + 560, 700} + 2260 = 3420 5. Example—application to AGVS guide path design
The algorithm developed is applied to design the guide path of AGV systems.
Parameters used in the algorithm are estimated as follows: (1) Required number of AGVS
To simplify the expressions, the following notation is used: 1396 K. H. Kim and J. M. A. Tanchoco 7}," = trip time (min) of route from station I to station m (route (I, m))
D1," =trip distance (feet) of route (I, m) S = average travel speed (feet per min, FPM) L=load/unload time (min) per trip E = trip inefﬁciency (min) which includes idle time, battery changing time and maintenance time per trip ﬁm= trip frequency per shift on route (I, m) U: available operational time per shift Then, total trip time of route (I, m) becomes
Tzm=Dim/S+L+E Then the number of AGVs required for route (l,m) becomes (21) initial layout P : Pickup station L: Load station — : Candidate aisles for
D : Delivery station U : Unload station AGVS lanes (b) ﬂow requirements Travel freq. per
shift 45
30 Figure 2. The initial layout and ﬂow requirements of example 1. Economical design of material ﬂow paths 1397 (a) initial layout 13 PD7/11 PD12 PDIJ PDM O—lT«I I I—I 30 at: so an 25 2!
PD] PD: P133 P D3 P09 g 25 a I
O m I 31 I” I, IT IT: —” 0— PD!!! AR/P : AS/RS pickup station
AR/D '. AS/RS delivery station PD : P/D station for work centers
— : Candidate aisle for AGVS lanes (b) ﬂow requirements l
AR/P PDl — PDS PDl6 AR/D
product2 AR/P— PD4— PD6 PD7  PDll  AR/D Material 50
40 Figure 3. The initial layout and ﬂow requirements of example 2. flmTl'm/U=flm(Dlm/S+L+E)/U
Finally, we get the total number of AGVs required, N=Zflm(D,m/S+L+E)/U (16) AGV related cost
(2) 0 C1 =AGV price
AGV cost: C1 x (number of AGVs)
o C2=batery/charger expense per vehicle
battery/charger cost: C2 x (number of AGVs)
0 C3 =maintenance cost (material and personnel) per vehicle per year 1398 K. H. Kim and J. M. A. Tanchoco (21) initial layout P/D : Part P/D Stalion TE'I‘ool P/D Station TM : Tool management station LU : Load/Unload Station C : Chip hopper P/D Station — : Candidate aisles for AGV lanes (b) flow requirements Travel freq. per shift
product 1 product 2 LU ~ PDZ  D4/P4  D3/P3  LU
product 3 LU  Dl/Pl  D3/P3  U
product 4 LU  PDZ  D3/P3  LU TM134  Tl  TM134 Figure 4. The initial layout and ﬂow requirements of example 3. (3) Guidepath and controls
0 C4=main guidewire cost per foot (including returns, trafﬁc controls, line
drivers, electric hookup)
0 C5 =Merges, stops, diverges, and intersections cost per path segment
0 C6 =space cost per foot per year (land cost, building cost, etc.)
(4) Others
0 C7: System controller, communication links, remote control unit, etc. Note
that the cost of controls associated with the guide path segments, intersec
tions and merges, etc. are included in C5. Economical design of material ﬂow paths 1399 (a) the ease with cost criteria
P1 P2 P3
[—_..  _> ____.... . .—p i r" 7‘” I u In
In“ ‘9'” l N (b) the case with travel distance criteria P1 P2 P3
'——>—>———F.——F i 7‘" T” I u .92
inlﬂ me
4 m D5
4—.4— ‘_—."—— 1 P6
D1! PB . ~1——1—4————? IN N... P : Pickup station L : Load station —> : Lane to be constructed
D : Delivery station U : Unload station _ : Space available but no lane
to be constructed Figure 5. The optimal ﬂow path for example 1. 0 C8: Maintenance cost per year excluding costs for vehicles. Although some
kind of maintenance cost is dependent on the ﬂow path design, it is assumed
to be constant considering the difﬁculties involved in obtaining its valuation. The total cost per year (TC) may be expressed as
TC=rNC1 +rNC2+NC3+rZC4dijZij+rZC5ZU+ZCédijzij+rC7+C8 (17)
L} 111' H where r is a factor which transforms the initial investment into the equivalent annual
cost. Replacing N using equation (16) and rearranging the terms 1400 K. H. Kim and J. M. A. Tanchoco rC1 +rC2+C3 TC= U Zﬁm(Dlm/S+L +E)+Z(rC4d,j+rC5+Cédij)Z,j=rC7+C8
l,m ixj If we delete the terms independent of the design of ﬂow path, the total cost function
becomes rC1 l—rC2 + C3 TC]: ZflmZXijlmdi j+Z(rC4di j+rC5+C6dij)Zij l,m t',j (a) the case with cost criteria I'D‘II'I'I FBI2 I'D!) FDI‘ PDij PDIG
I 4——4—<——  T<__T =EE£=hE1£H=ZPDi=2PhPI=QQ+—>I FDIC! PEI! .PD6
AM‘ /
m . 22;}
AWE! (b) the case with distance criteria rn'm I PD] 2 pm: pma pm 5 ram
.‘,  .4...._...4_'h.‘__  4—4—_ £ Pm .im P08 I'm '
O ——" l—F'~‘ I—FIDm IbI—b 0+I—b I'D“) I‘DS I Put. arr—”WK ARA) : AS/RS Flak“P station PD : P/D station for work centers AR/D : AS/RS delivery station _> : Lane to be constructed — : Space available but no lane to be constructed Figure 6. The optimal ﬂow path for example 2. Economical design of material ﬂow paths 1401 (a) the case with cost criteria ﬁ ”' o
.ﬁ—Fﬁ __i
l: in i r3.2.2.322.» _. 1“
9 a
[—ﬁ . '—"' I ——F x
. 1'M134 T‘“ : PD:
.T' 1 c2
TM: P1 P3 f +
?+"—P "—FO—b $+g+g+C _>4_ u
1'
1 O “I": .12
i C3
I W c“
t D)
.+ . 41— 4.._— . 4—..— ‘—
D4
P/D :Part P/D Station T :Tool P/D Station TM : Tool management slnlion
LU : Load/Unload Stallon C : Chip hopper P/D Station ‘— :Lane to be constructed : Space nvnllnblc but no lane constructed Figure 7. The optimal ﬂow path for example 3. Number of lower Execution time (s) bounds evaluated
Optimal
To get To conﬁrm To get To conﬁrm objective
Example optimum optimality optimum optimality value
1 13 53 51 178 30 756
2 98 169 246 388 43 990
3 350 1639 325 1102 26 254 Table 1. Computational results for the examples with cost criteria. 1402 K. H. Kim and J. M. A. Tanchoco Figure 8. Networks used to generate test problems. Figure 9. An example of generated problems. Thus, if we set
_ rC1 + rCz + C3 WZW‘M US and
Kij=rC4dij+rC5+C6dij then, the total cost function coincides with the objective function (1).
In the numerical example, we use the following data: Economical design of material ﬂow paths 1403 c,=$52000 C2=$4800 C3=$3100
C4=$20 C5=$1400 C6=$16
S=150FPM U=480minr=0~1 Then, W= Wm=012 and K“: l8d,J+ 140. Using the three layouts of real systems shown in Figs 2(a), 3(a), and 4(a) where
AGVs are being used in material handling, the algorithm developed is applied to
design the minimum cost ﬂow "path. The ﬂow requirements derived from the process
plan and the production amount of products are also contained in Figs 2(b), 3(b),
and 4(b). The optimal ﬂow paths obtained by the algorithm developed are shown in
Figs 5(a), 6(a), and 7(a). Note that some aisles are not used even though the spaces
are available. Computational results are summarized in Table 1. 5.1. Special case—travel distance criterion Note that the model developed by Kaspi and Tanchoco is a special case of the
model in this paper which has the single lane restrictions for all aisles and WW: 1,
Kij=0 for all I, m, i and j. Thus, unidirectional AGV guide path design problems
may be solved using the model presented in this paper. The solutions for the special
case where travel distance is the only criterion used are illustrated in Figs 5(b), 6(b),
and 7(b) for the problems with the same initial networks as the three examples in the
cost model. It is noted that the optimal ﬂow path networks are quite different for the
cost model and the travel distance model. 6. Computational performance To evalute the computational performances of the algorithm, the computation
times were compared with the algorithm developed by Kaspi and Tanchoco for the
problems whose objective function is to minimize the total travel distance without
considering the ﬁxed cost for lanes. The same three examples for cost model, one example from Kaspi and Tancho
co’s paper and 15 problems which are generated randomly, were solved using both
algorithms. The results are summarized in Table 2. Although the performances were
similar for small problems, the differences in computation time become more
signiﬁcant as the size of problems becomes larger. The results of computational
experiments show that the execution times of Kaspi and Tanchoco’s algorithm
exceed those of this paper for all the problems with 10 or more intersections. All computational comparisons were performed on a SUN workstation 4/390.
The program, ﬂow path generator (FlowPathGen), used to generate 15 test
problems is listed in the appendix. An example of input data to the program and
ﬂow path networks used for test problems are also provided in the appendix and
Figs 8 and 9, respectively. 7. Conclusions A branch—andbound procedure is suggested to design a ﬂow path for ﬁxed path
material handling systems. A cost model is constructed which considers the
construction cost, space cost and cost for controls for each path segment as well as
the travel cost. A tighter lower bound than the one of Kaspi and Tanchoco is derived
and the depthﬁrst searching strategy is adopted to get a good upper bound in the 1404 K. H. Kim and J. M. A. Tanchoco Execution time (s)
No. of
NO. of ﬂow Optimal Kaspii Algorithm
No. of intersec require objective Tanchoco’s in this Problem nodes tions ments value algorithm paper
Fig. 1(b) 21 5 21 128 735 7 13
Fig. 2(b) 23 6 25 204 005 13 12
Fig. 3(b) 30 10 27 79 585 401 54
Example from Kaspi~Tanchoco 23 14 22 10 170 765 226
Generated problems 18 8 12 — 11* 15*
Generated problems 25 12 18 — 183* 100*
Generated problems 32 16 24 7 ** 543* *These values are the averages of ﬁve generated problems.
**Four among ﬁve problems were not solved within 2000 s which was given as the time
limit, while one problem was solved in 1189 s. Table 2. Computational results for problems with travel distance criteria. early stage. Constraints on the directions of arcs in intersections are utilized to speed
up the branching process and reduce the number of computations of the lower
bounds (Sinriech and Tanchoco 1991). Three real layouts are used to illustrate the algorithm developed. And the
computational performances are compared with the algorithm of Kaspi and
Tanchoco, which dealt with a special case of the problem assumed in this paper.
Nineteen problems including 15 problems generated randomly are solved. The
results of comparison show that the algorithm of this paper outperforms Kaspi and
Tanchoco’s algorithm signiﬁcantly for largesized problems. The algorithm suggested in this paper may be applied to design automated
monorail systems, automated guided vehicle systems, automated guided vehicle
systems, ﬂexible conveyor systems, and other ﬁxed path material handling systems,
where the conﬁguration of the ﬂow path inﬂuences the performance of the whole
material handling system signiﬁcantly. Appendix. Program to generate the test problems
CC*4!******************)k****************************** CC * FlowPathGen4F low Path Generator Program *
CC 1‘ *
CC * by *
CC * *
CC * Kap Hwan Kim *
CC * Pusan National University *
CC * 4!
CC * and *
CC * *
CC * J.M.A. Tanchoco *
CC * Purdue University *
CC * *
CC ******it******i*******1K******************************** DIMENSION D(50,50),Q(50,50),NS(50,6),NQR(50),NQR(50),
1NSS(50,6)
COMMON SEED
CC NN:NUMBER OF NODES OF INPUT NETWORK
CC ND:NUMBER OF DEPARTMENTS Economical design of material ﬂow paths 1405 cc NSS(I,J): NODES SURROUNDING DEPARTMENT 1 INDEXED 1N CLOCKWISE ORDER CC LEDGEzLENGTH OF EDGE CC NREQzNUMBER OF NETWORKS REQUIRED cc QBEGzBEGINNING AMOUNT OF FLOW REQUIREMENT cc QSTEP:STEP SIZE OF FLOW REQUIREMENT
SEED = 7.
READ(5,*)NN,ND,NREQ,LEDGE,QBEG,QSTEP
NN=NN+ND
DO 1 1=1,NN
DO 1 J=1,NN
Q(LJ)=0 1 D(I,J)=10.**10.
DO 211=1,ND 2 READ(5,*)I,NSS(I,1),NSS(I,2),NSS(I,3),NSS(I,4)
DO 4 11=1,ND 4 NSS(Il,5)=NSS(Il,1)
DO 1000 KKK: 1,NREQ
WRITE(6,3005)KKK 3005 FORMAT(///1X,19HTHIS IS THE PROBLEM,1X,12)
DO 99911=1,NN
DO 9991J=1,NN
Q(I,J)=0. 9991 D(I,J)=10.**10.
DO 9992 11 =1,ND
DO 9992 11:1,5 9992 NS(II,J1)=NSS(II,J1)
DO 3 11 = 1,ND
DO 3 J1 = 1,4
IF(D(NS(Il,J1),NS(Il,J1 + 1)).NE.LEDGE.AND.D(NS(11,Jl + 1) 1,NS(Il,Jl)).NE.LEDGE)THEN D(NS(Il,Jl),NS(Il,Jl +1))= LEDGE
NNR=NRAND(1,4)
END IF 3 CONTINUE
DO 5 11 = 1,ND 6 NRAN=NRAND(1,4)
IF(D(NS(I1,NRAN),NS(II,NRAN+1)).EQ.10.**10..AND.D(NS(11, lNRAN+1),NS(Il,NRAN)).EQ.10.**10.)GO TO 6 D(NS(11,NRAN),NS(11,NMN+1))=1o.**10.
D(NS(Il,NRAN+1),NS(II,NRAN))=10.**10.
D(NS(I 1 ,NRAN),II) = LEDGE/2. + 2.
D(Il,NS(Il,NRAN +1))= LEDGE/2. +2.
KRAN=NRAND(1,4) 5 CONTINUE NQ=0 NR=0 NC=0 NTASS = 0 D0 7 11 = 1,ND NQR(11)=0 NQC(11)=0 1E(NR.GE.ND.AND.NC.GE.ND)GO TO 10 IRAN=NRAND(1,ND) JRAN=NRAND(1,ND) IF(NR.EQ.((ND — 1))THEN DO 543 154:1,ND IF(NQR(154).EQ.0)152=154 543 IF(NQC(IS4).EQ.0)ISI=IS4
IF(152.EQ.151)THEN 542 KRAN=NRAND(1,ND)
LRAN=NRAND(1,ND)
IF(KRAN.EQ.152.0R.LRAN.EQ.152)GO TO 542
NTASS =NTASS +1
Q(152,KRAN)=QBEG+(NTAss —1)*QSTEP
NQ=NQ+1 OO\O\I 1406 K. H. Kim and J. M. A. Tanchoco NTASS=NTASS+1
Q(LRAN,152)=QBEG+(NTASS—1)*QSTEP
NQ=NQ+1
GO TO 10
END IF
END IF
IF(NQR(IRAN).EQ.1.0R.NQC(JRAN).EQ.1)GO TO 8
IF(IRAN.EQ.JRAN)GO TO 8
NTASS=NTASS+I
Q(IRAN,JRAN) = QBEG + (NTASS — 1)*QSTEP
NQ=NQ+1
NQR(IRAN)=1
NQR(JRAN)=1
NR=NR+1
NC=NC+1
GO TO 9 10 MM =2*ND
IF(NTASS.GE.MM)GO TO 2000 11 IRAN=NRAND(1,ND)
JRAN=NRAND(1,ND)
IF(Q(IRAN,JRAN).NE.0.)GO TO 11
IF(IRAN.EQ.JRAN)GO TO 11
NTASS=NTASS+1
Q(IRAN.JRAN)=QBEG+(NTASS—1)*QSTEP
NQ=NQ+1 '
GO TO 10 2000 WRITE(6,3000) 3000 FORMAT(//1X,4HNODE,1X,4HNODE,1X,8HDISTANCE)
DO 12 I=1,NN
DO 12 J=1,NN
IF(D(I,J).GE.10.**10.)GO TO 12
WRITE(6,3001)I,J,D(I,J) 3001 FORMAT(1X,I4,1X,I4,1x,F8.1) 12 CONTINUE
WRITE(6,3002) 3002 FORMAT(//1X,4HNODE,1X,4HNODE,1X,16HFLOW REQUIREMENT)
DO 13 I=1,NN
DO 13 J=1,NN
IF(Q(I,J).EQ.0.)GO TO 13
WRITE(6,3003)I,J,Q(I,J) 3003 FORMAT(1X,I4,1X,I4,1X,F8.1) 13 CONTINUE 1000 CONTINUE
STOP
END
FUNCTION NRAND(I,J)
COMMON SEED
RR=RAND(SEED)
SEED = RR
NRAND=FLOAT(I)+FLOAT(J+1—I)*RR
RETURN
END Input data to 'the program for network of Fig. 8(a) 12,6,5,20,5.,5.
1,7,8,12,11
2,8,9,13,12
3,9,10,14,13
4,11,12,16,]5
5,12,13,17,l6
6,13,14,18,17 Economical design of material ﬂow paths 1407 References DIONNE, R., and FLORIAN, M., 1979, Exact and approximate algorithms for optimal network
design. Networks, 9, 37—59. FLOYD, R. W., 1962, Algorithm 97—shortest path. Communications of ACM, 5, 345. GALLO, G., 1983, Lower planes for the network design problem. Networks, 13, 411—425. GASKINS, R. J., and TANCHOCO, J. M. A., 1987, Flow path design for automated guided
vehicle systems. International Journal of Production Research, 25, 667~676. GASKINS, R. J ., TANCHOCO, J. M. A., and TAGHABONI, F., 1989, Virtual ﬂow paths for free
ranging automated guided vehicle systems. International Journal of Production
Research, 27, 91~100. GOETZ, W. G., and EGBELU, P. J ., 1990, Guide path design and location of load pickup/drop
off points for an automated guided vehicle system. International Journal of Production
Research, 28, 927—941. HOANG, H. H., 1973, A computational approach to the selection of an optimal network.
Management Science, 19, 488—498. KASPI, M., and TANCHOCO, J. M. A., 1990, Optimal ﬂow path design of unidirectional AGVS
systems. International Journal of Production Research, 28, 1023—1030. LOS, M., and LARDINOIS, C., 1982, Combinatorial programming; statistical optimization and
the optimal transportation network problem. Transportation Research, 16B, 89—124. MAGNANTI, T. L., and WONG, R. T., 1984, Network design and transportation planning:
models and algorithms. Transportation Science, 18, 155. RABENECK, C. W., USHER, J. S., and EVANS, G. W., 1989, An analytical model for AGVS
design. International Industrial Engineering Conference & Societies‘ Manufacturing and
Productivity Symposium Proceedings, 191—195. SINRIECH, D., and TANCHOCO, J. M. A., 1991, Intersection graph method for AGV ﬂow path
design. International Journal of Production Research, 29, 172571732. ...
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