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Design of materialtransportation system fortandem automated guidedvehicle systemsC. HuangPublished online: 14 Nov 2010.

To cite this article: C. Huang (1997): Design of material transportation systemfor tandem automated guided vehicle systems, International Journal ofProduction Research, 35:4, 943-953

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Design of material transportation system for tandem automated guidedvehicle systems

C. HUANG²

A new design concept of material transportation system is proposed on the basis oftandem automated guided vehicle (AGV) systems. The ® rst design stage is to ® ndthe one and only one optimal transfer point for each tra� c zone. All in-processparts to be moved into and out of a zone must go through the speci® ed transferpoint and parts can be moved directly to their destinations. The second designstage is to link all transfer points as a t̀ransportation centre’ by several bi-directional tracks. This design concept can simplify material transportationsystem and provides accurate estimation of tra� c load in a zone.

1. IntroductionThe tandem concept for AGV systems was ® rst introduced by Bozer and

Srinivasan (1989, 1991). As shown in Fig. 1, a tandem AGV system is obtained bypartitioning all workstations into single-vehicle, non-overlapping zones. A zone canbe de® ned as one and only one unidirectional AGV performs all moves within aspeci® c zone with the ®̀ rst-encountered-® rst-served’ dispatching rule. Additional pickup and deposit points (Maxwell and Muckstadt 1982) are provided between adjacentzones to serve as t̀ransfer points’ . Bozer and Srinivasan further develop a heuristicpartitioning scheme to con® gure tandem AGV system on the basis of a variation ofthe well-known set partitioning problem (Bozer and Srinivasan 1992). The advan-tages of the tandem concept are: ® rst, simpli® cation of the control system since onlyone AGV performs the materia l handling jobs within a tra� c zone independently.Second, the tra� c congestion and blocking problems can be eliminated . Third, itoŒers more ¯ exibility since zones or workstations within a zone can be added,removed, or modi® ed without aŒecting other zones (Bozer and Srinivasan 1992).However, several limitations and cost impacts of the tandem concept still exist asfollows (Chen 1995): ® rst, a load may have to be handled by several vehicles before itreaches its destination. For example, in Fig. 1, three AGVs and two pick-up/deposit(or transfer) points may be required to perform a single movement from zone 1 tozone 5. Consequently, the inter-zone tra� c control becomes more complicated .Second, additional ¯ oor space for pick up and deposit operations is required toprovide the transfer points. For example, in Fig. 1, four transfer points are requiredfor all pick up and deposit operations in zone 3. Additional installation cost andadditional facilities for pick up and deposit operations are also required in thissystem. Third, a lot of t̀hrough tra� c’ exists in the system, which makes theestimation of tra� c load for each AGV within a tra� c zone more complicated .For example, in Fig. 1, a movement from zone 2 to zone 5 must go through zone 3 orzone 4. In another words, the AGV in zone 3 or zone 4 should reserve some extra

0020±7543/97 $12.00 � 1997 Taylor & Francis Ltd.

INT. J. PROD. RES., 1997, VOL. 35, NO. 4, 943±953

Revision received April 1996.² Department of Industrial Management, National Yunlin Institute of Technology, 123

University Road, Section 3, Touliu, Yunlin, Taiwan 640, Republic of China.

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capacity for the through tra� c. From the above observations, it is clear that thematerial transportation between zones still can be improved or simpli® ed.

In this paper a new design concept of materia l transportation system for tandemAGV systems is proposed. The major idea of this design concept is to minimize thetotal number of transfer points. The approach to implement this design concept is toallow one and only one transfer point in each zone. A load being moved into and outof a zone must go through a pre-de® ned transfer point of this zone. This load is thenhandled by the AGV belonging to a t̀ransportation centre’ . Conceptually, atransportation centre consists of several isolated AGVs and several bi-directiona ltracks which link all transfer points in the system. A transportation centre performsall the materia l handling jobs between zones (or between transfer points) indepen-dently. The most important feature of this approach is each load being handleddirectly to its destination by a single AGV only. A typical example of this designapproach is illustrated in Fig. 2. The bold lines in Fig. 2 indicate the transportationcentre of this tandem AGV system. This design approach oŒers three principaladvantages. First, additional ¯ oor space for pick up and deposit operations is notrequired and the additional cost for facilities in pick up and deposit points is alsoeliminated . Second, the materia l movements between zones are performed e� cientlyand independently. Third, the estimation of tra� c load within a single zone can bemore accurate since extra capacity for through tra� c does not need to be reserved.In the previous research work, Sinriech and Tanchoco (1991) developed animproved branch-and-bound algorithm for the ¯ ow path design in a conventionalnetwork AGV system; Lin et al. (1994) developed a two-phase approach to solve the

944 C. Huang

Figure 1. A typical tandem AGV system.

F igure 2. The proposed material transportation system for a tandem AGV system.

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load-routeing problem in a tandem AGV system. The basic assumption in theirresearches is the ® xed and known locations of all transfer points. However, the ® rstdesign stage proposed in this paper will focus on ® nding the optimal location oftransfer point for each zone.

Two important issues of this design approach should be further developed in thefollowing two sections. Once all workstations have been partitioned into non-over-lapping zones, the ® rst issue of this design approach is to ® nd the optimal location forthe unique transfer point in each zone, which will be discussed in §2. This section willformulate the problem ® rst, and then derive a general solution procedure to reducethe solution space. Based on the solution procedure developed, the optimal locationcan be easily found by checking the limited feasible solutions from the reducedsolution space. After the locations of all transfer points have been found, the secondissue is to construct a transportation centre, which will be developed in §3. An integerprogramming approach is used to formulate this problem and the solution algorithmof the minimal spanning tree by Kruskal (Hiller and Lieberman 1990) is suggested forsolving this problem. In addition, a numerical example for ® nding the optimallocation of a speci® c tra� c zone and constructing a transportation centre is givenin §4. The concluding comments of this paper are in the last section.

2. The optimal location for a single transfer pointThe decision of the optimal location for the single transfer point is performed

when all workstations have been partitioned into zones, which means that the layoutof the tandem AGV system is ® xed and known. The following developments arebased on four assumptions: ® rst, the rectilinear moving distance for all loads in thesystem and second, only one moving direction (one-way movement) for an AGVwithin a tra� c zone. Third, one unit load requires a single movement. Fourth, thefeasible locations of a transfer point can be assigned at any point along the movingtrack of a speci® c zone excluding the points where workstations have been located.

A typical tra� c zone with a closed-loop track is presented in Fig. 3. There are nworkstations in this tra� c zone and the length of this closed-loop track is assumedto be L. All workstations are numbered in sequence, as P1, P2, . . . and Pn , starting

945A material transportation system for the tandem AGVS

Figure 3. A typical tra� c zone.

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from the moving direction of an original point (A) which is selected arbitrarily. Foranalysis purpose, the transfer point (t) should be linked to this closed-loop zone at apoint s by a bi-directional moving track (a ® xed length of this track is assumed to ber). It is possible that the point s can be located at any point between workstations, e.g.between workstation k and workstation k + 1 (1 £ k £ n). The objective function for® nding the optimal transfer point (t) is to minimize the total moving distances (z)between the transfer point t and workstations, which can be stated as equation (1).

Min . z = Sn

i=1

( fit dit + fti dti) (1)

Where: fit represents an aggregated ¯ ow quantity from workstation i to transferpoint t.fti represents an aggregated ¯ ow quantity from transfer point t to work-station i.dit is the distance from workstation i to transfer point t.dti is the distance from transfer point t to workstation i.

Expression fit also implies the materia l being moved out of this tra� c zone from aspeci® c workstation i. fti implies, on the other hand, the materia l being moved intothis tra� c zone. Since the one-directiona l movement is assumed within the tra� czone, dit and dti may not be the same. The relationship between dit and dti can berepresented as equation (2).

dit + dti = L + 2r (2)

Based on equation (2), the equation (1) can be modi® ed as equation (3).

z = Sn

i=1

( fit dit + fti (L + 2r - dit)) = Sn

i=1

( fit dit + fti (L + 2r) - fti dit)

= Sn

i=1

( fit - fti) dit + Sn

i=1

fti (L + 2r) (3)

In Fig. 3, if x represents the moving distance from the original point (A) to transferpoint (t), then dit can be expressed as one of the following two equations, i.e. equation(4) or (5).

dit = x - dAi ,

x + diA,

1 £ i £ k

k + 1 £ i £ n{(4)

(5)

Therefore, the equation (3) can be further modi® ed as follows.

z = Sn

i=1

( fit - fti) dit[ ]+ Sn

i=1

fti (L + 2r)[ ]= S

k

i=1

( fit - fti) dit + Sn

i=k+ 1

( fit - fti) dit[ ]+ Sn

i=1

fti (L + 2r)[ ]= S

k

i=1

( fit - fti) (x - dAi) + Sn

i=k+ 1

( fit - fti) (x + diA)[ ]+ Sn

i=1

fti (L + 2r)[ ]

946 C. HuangD

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= Sk

i=1

( fit - fti)x - Sk

i=1

( fit - fti)dAi + Sn

i=k+ 1

( fit - fti)x + Sn

i=k+ 1

( fit - fti)diA[ ]+ S

n

i=1

fti (L + 2r)[ ]= S

n

i=1

( fit - fti)x[ ]+ - Sk

i=1

( fit - fti)dAi + Sn

i=k+ 1

( fit - fti)diA[ ][ ]+ S

n

i=1

fti (L + 2r)[ ]= [G1(x) + G(k)]+ [G] (6)

= G(x) + G (7)

Where:

G1(x) = Sn

i=1

( fit - fti)x (8)

G(k) = - Sk

i=1

( fit - fti)dAi + Sn

i=k+ 1

( fit - fti)diA (9)

G = Sn

i=1

fti (L + 2r) (10)

G(x) = Sn

i=1

( fit - fti) dit = G1(x) + G(k) (11)

Based on the equation (10), G is a constant term, therefore, minimization of z can bereplaced by minimization of G(x) which is shown in equation (11). G(k) is a functionof k. If a workstation k is given in G(k), then G(k) becomes a constant term. G1(x) is alinear function of x and G1(x) and k are independent. Therefore, for any given k, G(x)is a linear function of x and the slope of this line (i.e. å n

i=1 ( fit - fti)) is also a ® xedvalue.

Since x is continuous between xk and xk+ 1 for any given k, the extreme values (i.e.the maximal and minimal values) of G(x) will occur at both ends of a line segment.For a single line segment at a given k, three situations may be concluded as follows:® rst, when å n

i=1 ( fit - fti) > 0, G(x) can be plotted as a line segment with a positiveslope as shown in Fig. 4. In this case, when x is increasing, the value of G(x) is alsoincreasing. The minimum of G(x) occurs at x

+k which is located at the output point of

the workstation k. The output point can be de® ned as the point as close to theworkstation k as possible on the output side of the workstation k. (In a real-worldsituation, a transfer point cannot be located at a workstation itself since it may causetra� c congestion problems as in the fourth assumption mentioned in §2.) Second,when å n

i=1 ( fit - fti) < 0, the minimum of G(x) occurs at x-k+ 1 which is located at the

input point of the workstation k + 1 as shown in Fig. 5. A similar de® nition can be

947A material transportation system for the tandem AGVSD

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made; the input point is the point as close to the workstation k + 1 as possible on theinput side of the workstation k + 1. Third, when å n

i=1 ( fit - fti) = 0, the minimum ofG(x) occurs at any point between x

+k and x

-k+ 1.

Based on the above analysis, the solution space of the optimal transfer point canbe e� ciently reduced from a line segment to several limited points (i.e. 2n points). Astepwise procedure for ® nding the optimal transfer point can be summarized asfollows.

Step 1. Calculate the unique slope of the line segment using å ni=1 ( fit - fti).

Step 2. Based on a ® xed original point (A), formulate G(x) for k = 1, 2, . . . , n usingequation (11).

Step 3. Calculate the values of G(x) using x at both ends of each line segment (i.e. x+k

and x-k+ 1, for k = 1, 2, . . . , n).

Step 4. Find the minimal G(x) from step 3 and denote the corresponding k as P *k.

Step 5. If å ni=1 ( fit - fti) > 0, then the optimal transfer point is located at P *+

k .If å n

i=1 ( fit - fti) < 0, then the optimal transfer point is located at P *-k+ 1.

If å ni=1 ( fit - fti) = 0, then the optimal transfer point is located at any point

between P *+k and P *-

k+ 1. (i.e. Min. G(x) = G(P *k) = G(P *

k+ 1)).

3. Construction of a transportation centreThe basic concept of the transportation centre is to independently perform all

the inter-zone movements directly to destinations. Several bi-directional (two-way)

948 C. Huang

Figure 4. Values of G(x) between xk and xk+ 1 (for a positive slope).

Figure 5. Values of G(x) between xk and xk+ 1 (for a negative slope).

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travelling tracks are required to link all transfer points in the system. Once the movingpaths between transfer points are con® rmed, the tra� c load of the transportationcentre is then calculated for estimation of the AGV requirement (Tanchoco et al.1987, Chen 1995). One, or more than one AGV may be assigned to a transportationcentre independently. The objective of constructing a transportation centre (orconstructing bi-directiona l tracks) is to minimize the total distances of tracks betweentransfer points so that all the inter-zone movements can be handled e� ciently.

In this section, the concept of a tree structure for linking all transfer points isproposed ® rst. Based on this concept, a mathemat ical model and the associatedsolution approach are suggested. A tree structure can be de® ned as follows. (1) If nnodes (transfer points) exist in the system, there will be n - 1 line segments to link allnodes. (Each line segment connects two nodes.) (2) There are at least two nodesexisting in a tree structure. (3) There is one and only one possible path between twonodes. (4) Two-way movements are allowed on all paths. The objective function andconstraints of a transportation centre with the tree structure concept can be indicatedas follows.

Min . z = Sn

i=1S

n

j=1

dij xij , i /= j (12)

s. t . Sn

i=1

(xij + xji) ³ 2, " j = 1, 2, . . . , n and j /= i (13)

Sn

i=1S

n

j=1

xij = 2(n - 1), i /= j (14)

xij - xji = 0, " i, j and i /= j (15)

xij Π{0, 1} (16)

Where:

n is the number of transfer points in the system.xij = 1, (decision variable) If a track between transfer point i and transfer point j

exists,xij = 0, otherwise.dij is the distance between transfer point i and transfer point j and dij = dji.

The objective function is to minimize the total distances of tracks between transferpoints as indicated in equation (12). Equation (13) indicates that each transfer pointshould be linked to one or more than one other transfer points. Equation (14) showsn - 1 tracks existing in the system. Equation (15) indicates that two-way movementsare acceptable.

The minimal spanning tree is one of the solution algorithms for this type ofproblem proposed by Kruskal (Hiller and Lieberman 1990). In application , thisalgorithm can be summarized as the following steps.

Step 1. Choose one transfer point arbitrarily as the initial point. Link this initial pointto the transfer point with the shortest distance. A simple tree structure isformed.

Step 2. Select the transfer point having the shortest distance with the current treestructure. Link this transfer point to the current tree structure and form a newtree structure.

949A material transportation system for the tandem AGVSD

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Step 3. Repeat step 2 until all transfer points have been included in the current treestructure.

In the real-world situation, the length of two transfer points may or may not be arectilinear distance. A shorter distance such as a Euclidean distance, or a longerdistance for avoiding tra� c congestion, may occur in real-world situations. Thisdrawback indicates that the consideration of moving distance must cope with the real-world constraints and limitations.

4. A numerical exampleIn this section, a hypothetical example is given to demonstrate and explain the

stepwise procedures proposed in §2 and §3 respectively. The layout of this tandemAGV system with two tra� c zones is shown in Fig. 6. The ¯ ow quantities betweenworkstations are also given in Table 1. Zone 1 consists of three workstations (i.e. P2,P2, and P3) and the rest of the workstations are included in zone 2 (i.e. P4, P5, and P6).The ® rst design stage is to ® nd the optimal location of a transfer point for each zone.The explanation of this stage will concentrate on zone 1 and the same procedure canbe used in zone 2. In zone 1, P1 is chosen as the original point (A) and the fit and fti

can be calculated as follows.

950 C. Huang

Figure 6. An example of tra� c zones (workstations, moving tracks, and moving direction).

TO

P1 P2 P3 P4 P5 P6

P1 0 6 13 3 10 7F P2 15 0 7 8 16 15R P3 12 7 0 11 5 9O P4 10 2 7 0 8 3M P5 8 3 5 9 0 2

P6 10 15 20 7 1 0

Table 1. F low quantities between workstations.

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Þf1t = f14 + f15 + f16 = 20

f2t = f24 + f25 + f26 = 39

f3t = f34 + f35 + f36 = 25

ft1 = f41 + f51 + f61 = 28

ft2 = f42 + f52 + f62 = 20

ft3 = f43 + f53 + f63 = 32

ìïíïî

ìïíïî

fit = f1t + f2t + f3t = 84, fti = ft1 + ft2 + ft3 = 80

For each k (i.e. k = 1, 2, and 3 in this case), G(x) is a linear function of x. Threelinear equations are formulated in the seventh column of Table 2 and they are plottedas three line segments with the same slope (i.e. fit - fti = 4), as presented in Fig. 7.Once the slope and three linear equations are known, the solution space is eŒectivelyreduced to three points only. In this case, only P

+1 , P

+2 , and P

+3 need to be further

951A material transportation system for the tandem AGVS

Possible Locations for Calculation Calculation ofTransfer Point of dit Formulation of G(x) G(x)

Between At both ends:k 2 WSs Range of x d1t d2t d3t (x+

k , x-k+ 1)Using S

n

i=1

( fit - fti) dit( )1 (P1, P2) 0 < x < 7 x 7 + x 3 + x ( - 8)x + 19(7 + x) (112, 140)

+ ( - 7)(3 + x) = 4x + 112

2 (P **2 , P3) 7** < x < 11 x x - 7 3 + x ( - 8)x + 19(x - 7) ( - 126*, - 110)

+ ( - 7)(3 + x) = 4x - 154

3 (P3, P1) 11 < x < 14 x x - 7 x - 11 ( - 8)x + 19(x - 7) (- 12, 0)+ ( - 7)(x - 11) = 4x - 56

Remarks*Indicates the minimal (or the optimal) value of G(x) = - 126.**Indicates the optimal location can be installed at the output point of workstation 2. The

optimal transfer point is denoted as P *+k = P*+

2 = x+2 = 7

+.

Table 2. Formulation and Calculation of G(x): (assume P1 as the original point).

F igure 7. Relationship between G(x) and x (assume P1 as the original point).

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selected since the positive slope is observed. One of these output points will be theoptimal location for the transfer point. The minimum of G(x) is - 126 which is shownin the last column of Table 2. The optimal location for the transfer point of zone 1 isP*+

2 (i.e. x+2 = 7

+) which is the output point of workstation 2, as indicated in Fig. 7.

The same approach can be applied in zone 2 and the optimal location for the transferpoint is P *-

5 (i.e. x-5 = 12

-) which is the input point of workstation 5.

Once the locations of all transfer points have been found, the second design stageis to link all transfer points as a transportation centre with several bi-directiona ltracks. Using the algorithm suggested in §3, the ® nal transportation centre isconstructed by a simple tree structure, as shown in Fig. 8, since only two transferpoints should be linked. However, the distance of this transportation centre (i.e. closeto 5 + 6r units of distance) is longer than either the rectilinear distance or Euclideandistance of two transfer points. This situation may occur in some real-world casessince it is assumed that the track of the transportation centre cannot intersect with anyzone in the system. In addition, this situation also implies that other assumptions orsolution algorithms can be investigated and compared for solving this type ofproblem. However, the algorithm suggested in this paper does screen out severalunfeasible paths on the basis of total moving distance standpoint.

5. ConclusionIn this paper, a new design concept of materia l transportation system for tandem

AGV systems is presented. A single transfer point for each tra� c zone provides asimple tra� c control for the inter-zone materia l movements. The concept of thetransportation centre can eŒectively and e� ciently reduce over-investment on facil-ities and shop ¯ oor space. However, the following related topics still can be furtherinvestigated.

(1) Two-way movements within a tra� c zone can also be considered in ® nding theoptimal location for a transfer point.

(2) Both the Euclidean distance and rectilinear distance can take into considera-tions simultaneously when we are modelling a transportation centre.

952 C. Huang

Figure 8. The ® nal result of the example problem.

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(3) The minimal spanning tree algorithm and the other related solution algo-rithms can be further investigated and compared.

AcknowledgementsThis research was supported by the National Science Council of Republic of

China under Grant no. NSC 85-2213-E-224-011.

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CHEN , W. L., 1995, A planning model for the tandem automated guided vehicle system. MSthesis, National Yunlin Institute of Technology, Taiwan, ROC.

H ILLER , F . S., and LIEBERMAN , G . J. (eds), 1990, Introduction to Operations Research, 5th edn(New York: McGraw-Hill).

LIN , J. T., CHANG , C. C. K., and LIU , W., 1994, A load-routeing problem in a tandem-con® guration automated guided-vehicle system. International Journal of ProductionResearch, 32, 411±427.

MAXWELL , W. L., and MUCKSTADT, J. A., 1982, Design of automatic guided vehicle system. IIETransactions, 14, 114±124.

SINRIECH, D., and TANCHOCO, J. M. A., 1991, Intersection graph method for AGV ¯ ow pathdesign. International Journal of Production Research, 29, 1725±1732.

TANCHOCO, J. M. A., EGBELU, P. J., and TAGH ABONI, F ., 1987, Determination of the totalnumber of vehicles in an AGV-based material transport system. M aterial Flow, 4, 33±51.

953A material transportation system for the tandem AGVSD

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