Flow_path_design_for_automated_guided_vehicle_systems

Flow_path_design_for_automated_guided_vehicle_systems -...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

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. TANCHOC-Oi‘ 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 direct-ions. 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 wit-h 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 zero-one 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 depart-mental 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. Pat-h 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 stat-ions, 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- sect-ions 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 pat-h design for AG V8 (in!) 10 (a) Figure l. (a) Example layout. {b} Corresponding net-work 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 ail-J- : 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 zero-one 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. Ta-nchoco 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 bot-h 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 non-linear 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 bet-ween 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 constraint-s 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 net-work 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. Net-work 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 pat-h 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 f01'22-node problem. (it - 2)(first are} + {a — 2)(last are) — (sum of the intermediate ares) 5-H (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- st-raint 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. Tam-hose 4.2. Parameters dmp distance from node-m 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 four-way nodes is given by Z I”. ; 2 j’ = is a 4-way 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 pat-h 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 from-to 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 (AG-VS: Automated Guided Vehicle Systems} est I’et-ude du circuit. d'et-oulement. 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 zero-1m. Des exemples sont fournis a l’appui. Die Projektierung der Fahrbahn stellt einen wichtigen Faktor bei dem Entwurf automatisch gelenkter F‘ahrzeugsysteme dar. Dieser Beit-rag stellt. cine Met-bode 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- aer Polytechnic Institute, U.S.A., November. BRADLEY, S. P., Hax. A. C.._ and MAGNANTI, T. L., 1977, Applied Mathemaiiml Program— 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 automatic guided vehicle based material handling systems. Technical report No. 8201, Department of Industrial Engineering and Operations Research. Virginia. Polytechnic Institute and State University, Blacksburg, Virginia, USA. Renew. P. J.. and Taxonoco, .J. M. A, Characterization of automatic guided vehicle dispatching rules. International Journal of Production Research, 22, 359—374. humus, R. L.._ and WHITE, J. A., 1974, Facility Layout and Location—An Analytical 4 A pin-roach {Englewood Cliffs, NJ: Prentice-Hall]. Human, F. 8., and LIEBERMANN, G. J., 1980, Introduction to Operations Research (San Francisco, CA: Holden—Day). .\lAX\\'II‘.I.I.. W. L. and li'll'trks'r‘nn'l‘. J. .-'\,. l982. Design of automatic guided vehicle systems. LLE. Transactions, 14, 114—124. MITIJLER, T .. 1983, Automated Guided Vela-totes (Bedford; IFS Publications Ltd). TANG-11000, J. M. A., EGBELL‘. P. J.. and TAGHABON], F., 1986, Determination of the total number of vehicles in an AGV—based material transport system. Mate-rial Flow (forthcoming). WATTERS. L. J.._ 1967. Reduction of integer polynomial problems to zero—one linear prog— ramming problems. Operations Research, 15, 1171—1174. ...
View Full Document

Page1 / 10

Flow_path_design_for_automated_guided_vehicle_systems -...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online