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Retrieval Strategies for a Carousel ConveyorJohn J. Bartholdi III a & Loren K. Platzman aa Georgia Institute of Technology School of Industrial and Systems Engineering , Atlanta,GA, 30332 1-404-894-3036Published online: 09 Jul 2007.

To cite this article: John J. Bartholdi III & Loren K. Platzman (1986) Retrieval Strategies for a Carousel Conveyor, IIETransactions, 18:2, 166-173, DOI: 10.1080/07408178608975344

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Retrieval Strategies for a Carousel Conveyor

JOHN J. BARTHOLDI, I11

and

LOREN K. PLATZMAN Georgia Institute of Technology

School of Industrial and Systems Engineering Atlanta, GA 30332

1-404-894-3036

Abstract: We analyze algorithms that sequence the retrieval of items f rom a carousel conveyor, and show how the appropriate algorithm depends o n the load to which the carousel is subjected. I n general, as the load increases, so does the quality of solutions produced by simple heuristics. We conclude by recommend- ing certain greedy heuristics for the problems of sequencing the retrieval of orders and for sequencing the retrieval of the items within each order.

A carousel conveyor is a length of shelf fashioned into a closed loop that is rotatable (under computer control) in either direction. It is increasingly found in automated warehouses since it can help considerably with the storage and retrieval of small parts. Among the benefits of a carousel is that, rather than have a picker, human or ro- botic, travel to retrieve an item, the item can travel to the picker. While the item is travelling, the picker has time to package, label, or pick from another carousel. This parallelism of activity enables greater throughput and therefore enhances the operational efficiency of the warehouse.

We consider a single carousel serviced by a single pick- er. The picker is to retrieve a fixed and known collection of items. We would like to do this as quickly as possible, on the premise that this will improve service and reduce cost.

An order is a set of items that must be picked together (as, f o ~ instance, for a single customer). We assume that only one order can be picked at a time, and that we must pick many orders. Our problem is to determine a sequence in which to pick the orders and the items within each ord- er so that the orders are retrieved as quickly as possible. We will propose several algorithms (computational proce-

Received November 1984: revised August 1985 and December 1985. Handled by the Scheduling/Planning/Control Department.

This research was supported by the National Science Foundation un- der grant #ECS-8351313, the Office of Naval Research under grant # N00014-80-k-0790, and by U.S. industry through The Material Han- dling Research Center at Georgia Institute of Technology.

dures) to solve this problem. The total time to retrieve all of the items may be ex-

pressed as a sum of two components: (I) the total time during which the carousel is travelling and (2) the total time during which the carousel is stopped for picking. For simplicity wc assume that the time to pick an item is negligible by comparison to the time to travel to thc item, and that the time to pick any item is independent of the sequence in which the items are picked. Thus, to shorten the total retrieval time we must shorten the time during which the carousel travels between storage loca- tions. We also assume that the carousel travels at a constant velocity, so that travel time is proportional to distance. Each item is stored at a single specified loca- tion on the carousel and may not be relocated during retrieval.

We model the carousel as a circle upon which are labelled points corresponding to storage locations (Figure 1). To avoid tortured grammar, we will occasionally speak of the picker as travelling around a stationary carousel, even though the opposite is true. Also it will be more convenient to speak of travel distances in place of travel times. Thus our problem is to determine a path of minimum length around the circle (with occasional reversals of direction), that visits each item to be picked, one order at a time.

Two measures of an algorithm are the quality of the solutions it produces and the speed with which it produces them. The quality of solutions produced by the al- gorithms to be discussed here will be judged by the total travel time, as defined above, or alternatively, the differ-

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Figure 1: The carousel is represented by the large circle and the storage locationsof itemsto be picked are small circles in thesame relative position as the corresponding items on the carousel. The pick station is indicated by the triangle. Rotating the carousel to bring items to the pick station corresponds to tracing a path from the pick station among the storage location of the items.

ence between the travel time produced by the algorithm and the best travel time time possible. For each algorithm the computation to produce a solution will consist of two parts. First, each order must be "preprocessed" so that the items to be picked are sorted according to storage lo- cation on the carousel, for example, starting at noon on the circle and proceding clockwise to midnight. Such son- ing tasks are easily programmed on a computer, and re- quire at least n log, n operations, where n is the number of items in an order. For each algorithm we propose, the remaining steps will require computational effort that is only linear in the number of items to be picked. There- fore, the computational effort is dominated by sorting, which is negligible. Thus the algorithms we propose run extremely quickly. Many are simple enough to be im- plemented manually, i.e., without a computer.

We are not aurare of any previously published work on control algorithms for carousel conveyors. The familiar model most expressive of these issues is the "travelling salesman problem" [3], which is to visit a number of locations and return to the starting point as quickly as possible. In our case the problem is posed on a relatively simple topology (the circle), but with the com- plications of sequencing both orders and the items within each order.

Item-density and Order-rate

How to best retrieve a collection of orders is a two- level sequencing problem: how to sequence items within an order, and how to sequence orders. These levels can-

not be entirely separated because the time to retrieve the next order depends on where the carousel finished retriev- ing the last one. To help understand how these two lev- els interact, at each level a parameter may be defined to express the load to which the carousel is subjected. The order-rate is the average number of orders received per unit time: the item-density is the average number of items within an order. We define these rigorously only to or- ganize our analysis; their exact values are unimportant.

Both the order-rate and the item-density affect how we might choose to solve the problems of sequencing orders and items. To help understand these effects we consider the extreme situations for each parameter.

(1) The order rate is small relative to the speed of order retrieval. If the order rate is sufficiently small, then the average time to retrieve an order is smaller than the aver- age time between the arrival of orders. Usually the carousel will retrieve the items of an order and then wait for the next order to arrive: consequently, orders will be retrieved in first-come, first-served sequence. Thus our retrieval problem decomposes into a series of problems, each of which is to quickly retrieve a single order.

(2) The order rate is large relative to the speed of ord- er retrieval, If the order rate is sufficiently high, the problem does not decompose, since new orders will ar- rive during the retrieval of other orders. Thus some ord- ers must wait during the unproductive time the carousel spends travelling from the end of one order to the begin- ning of the next, and we would like to minimize this wait- ing time.

The extreme situations for item-density are as follows. (1) The item-density is large relative to number of

storage locations in the carousel. If the item-density is surficiently large (say greater than 40% of the storage lo- cations), then to retrieve all the items of an order, the carousel must rotate past most of the storage locations. Such retrieval problems are mathematically easy since an effective (i.e. close to minimal time) route visiting all of the required storage locations is easily stated: simply ro- tate the carousel clockwise and pick items as they are en- countered until all items have been picked!

(2) The item-density issmaNrelafive to the number of storage locations in the carousel. In this case each retrieval of an order requires the solution of a routing problem which visits only a small number (say less than 10%) of the storage locations. The solutions to such problems can be very different, and so not amenable to the simple ap- proach above. In this case we must exercise ingenuity to quickly find effective retrieval routes.

The choice of retrieval strategy will therefore depend on the item-density and order-rate. When the order-rate is small, retrieval strategies can be simpler, since the problem decomposes into a succession of independent retrieval problems; when the order-rate is large, retrieval strategies must account for the sequencing of orders. When the item-density is small, retrieval strategies must be more complex in order to search for a quick picking

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sequence: when it is high, the retrieval strategies can he very simple and still produce near optimal picking se- quences.

Retrieval Strategies

When the Order-rate is Low In this case the retrieval problem decomposes into a

series of independent problems, each of which is to quickly retrieve one order. Thus we concentrate on how to sequence the retrievals of items within a single order. We first give an algorithm that will always find a short- est retrieval sequence. Then we discuss three alternative algorithms that are progressively simpler, but are poten- tially less accurate (i.e. can produce sequences that are increasingly longer than the minimum). As item-density increases, however, the successively simpler algorithms will perform adequately.

First we show that an optimal pick sequence can be found quickly. This is based on the observation that in retrieving a set of items from a carousel, it is never optimal to rotate the conveyor through a complete revo- lution since the additional travel from the last location back to the start is clearly wasted. Similarly it is never optimal to rotate past the same location more than twice since any such additional travel is excessive: consider a segment of the carousel that is traversed 3 or more times in some retrieval sequence: then a quicker sequence of retrievals is to replace that portion of the retrieval path by a path that traverses the same segment of the carousel only once or twice, and retrieves the same items. Thus some storage location (or else the current position of the picker) will be the "rightmost" in an optimal retrieval sequence. Now a storage location can be rightmost in at most two different ways: as in Figure 2, either the carousel can rotate counter-clockwise and then reverse direction to bring the item last to the picker, or the carousel can rotate clockwise to the item and then reverse direction. The starting location of the picker can be rightmost in only one way: if the carousel rotates counter-clockwise. Since each specification of a rightmost item determines 2 retrieval sequences for each of the n items to be retrieved, and there is one more sequence corresponding to the location of the picker, there are only 2n + 1 retrieval sequences that are candidates for optimality. Thus we have proved:

THEOREM 1: The following algorithmfinds the quick- est sequence in which to retrieve the items of a single order:

"Optimal Retrieval" Algorithm

Step 1: Evaluate the 2n + I candidate retrieval sequences that correspond to specifying a "rightmost" lo- cation.

Step 2: Choose the best of thc candidate solutions from Step 1.

Figure 2. The two different candidate retrieval sequences in which the marked storage location (x) is rightmost as seen from the current position of the picker.

This algorithm determines the quickest sequence in which to retrieve the items of an order. It requires only O(n) steps beyond the preprocessing sort. (We adopt the "big 0" notation of computer science to indicate that this al- gorithm runs in time that increases only linearly i nn , the number of items to be retrieved. In general such an al- gorithm is to be preferred to one that can require, for example, O(nZ) computational effort. See [I] for a care- ful definition of "big O".)

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The intelligence of this algorithm lies in its determin- ing the best place at which to reverse the direction of trav- el of the carousel. However, when the item-density is sufficiently large, it is unlikely that the optimal solution will double back (Imagine the extreme case in which item- density is so large that all storage locations mmt be visit- ed). This suggests that for larger item-densities, heuris- tics (simpler, non-optimal procedures) may produce adequate retrieval sequences. Here are three such heuristics.

"Nearest Item" Heuristic

Step 1: Always rotate to the nearest item to be retrieved.

This heuristic is simpler than the optimal algorithm. It is not guaranteed to produce the shortest retrieval route, but it does have provable performance bounds that guarantee the quality of its solutions.

THEOREM 2: Under Nearest Item, the total distance travelled to retrieve one order is never greater than one revolution of the carousel. Furthermore, it is never greater than twice the optimal distance.

Proof: Consider the "route" produced by Nearest Item. (It is convenient to imagine the picker moving around a stationary carousel). The route is composed of segments of uninterrupted clockwise movement or counter- clockwise movement; assume there are k such segments. For the Ph segment, let I, be the length along the seg- ment to the first item retrieved on that segment; let R, be the length of the remainder of the segment. (See Figure 3.) Then, since the heuristic always selected the nearest item to pick next,

Expanding this gives

r, 2 C (I, + R,). I = I

Thus each I,, and I, in particular, is at least as large as the total distance that has preceded it, so that the total distance travelled is never more than 2*Ik + R,. But 2*1, + R, is never more than one revolution of the carousel, which proves that the value of the heuristic so- lution is never more than one revolution.

That the value of the heuristic solution is never more than twice optimal is obtained as follows. Let I be the shortest interval that spans all of the items in the order. The distance travelled under the optimal sequence must be at least as long as I. If I is greater than or equal to one-half a carousel revolution, then the claim holds, since the heuristic requires no more than one revolution. If I is less than one-half a carousel revolution, then

Figure 3. A portion of the path traced by the Nearest Item heuristic. Whenever the carousel reverses direction it must travel at least twice as far in the new direction as in the previous. Specifically, 1. >

I - 2qT1 + Rj.,. The small circles represent storage locations from whlch items were picked.

I 2 I, + R,, and 2*(the distance travelled under the op- timal sequence) 2 2*I 2 2*(Ik + R,) r 2*1, + R, 2 the distance travelled under the heuristic.

When the item-density is large, it is unlikely that an optimal route will double back. In this case we can safely restrict our consideration to only 2 of the 2n + 1 candi- date routes for optimality: the 2 routes that do not dou- ble back. If we choose the shorter of these 2, then we are very likely to have chosen the shortest route. This sug- gests the following heuristic:

"Shorler Direction" Heurstic

Step I: Evaluate the length of the route that simply ro- tates clockwise, and the length of the route that simply rotates counter-clockwise.

Step 2: Choose the shorter of the two routes from Step 1.

While this heuristic never produces a route longer than the length of the carousel, it can, in the worst case, produce a route that is many times longer than the short- est (by a factor of one-half the number of carousel loca- tions). Nevertheless, its routes are likely to be optimal when the item-density is sufficiently large.

When the item-density is large with respect to the num- ber of carousel locations, then almost any reasonable heuristic will produce routes only negligibly longer than optimal, on the average. Thus, in the interests of eco- nomical computatjon, we might as well adopt the sim- plest heuristic.

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"Monomaniacal" Heuristic

Step 1: Always rotate to the right and pick items as they are encountered.

Thus as the item-density increases, the qualities of so- lutions of the retrieval strategies become indistinguisha- ble (and optimal). We might as well use the simplest retrieval strategy that works adequately, given the item- density.

When the Order-rate is High When the order-rate is high, the order-retrieval

problems do not decompose and we must sequence ord- ers, as well as the items within the orders. A typical way to organize retrieval in this case is to solve a "rolling" problem, i.e., t o specify some collection of orders to be retrieved, and ignore any new arrivals while retrieving them. This converts the dynamic problem to a series of (easier) static problems. Unfortunately, to solve even the static problem optimally seems difficult. Nevertheless there exist fast heuristics that give solutions guaranteed to be close to the optimal.

Again we study a succession of algorithms that are progresively simpler, but with weaker performance guarantees. And again we suggest that for sufficiently large item-density the simpler heuristics will produce retrieval sequences of nearly shortest length.

A lower bound on the travel distance to pick an order is the length of the shortest interval (on the circle) con- taining all the items of the order. We call this the "mini- mum spanning interval". It can be found in O(n) steps by omitting the largest gap between neighboring items of the order. We distinguish the endpoints of this interval by referring to them as the "endpoints of the order". If there are m orders to be picked, then there are 2m end- points of orders.

Any way of picking the orders must visit the two end- points of each order and so must travel at least the lengths of the minimum spanning intervals. For simplicity, we will require that the picker always begin retrieving an ord- er at one of the endpoints of the order and finish at the other. After picking the last item of one order, the pick- er travels t o pick the first item of the next order. This restriction may prevent the picker from acting in the best way possible, but we will show that it does not signifi- cantly compromise performance.

Under this rule, the total distance to pick all orders con- sists of two parts, the distance travelled within orders (which equals the sum of the spanning interval lengths

and therefore does not depend on the order sequence) and the distance travelled between orders. To minimize the latter, we must sequence orders so that the finishing end- point of each order is close to the starting endpoint of the following order. In other words, we must arrange 2m + 1 points (the 2m order endpoints and the initial pick- er position) into m pairs, representing the travel between orders, so as to minimize the total distance between points in a pair. In graph theory, this is known as a minimum- cost maximal matching [ I , 21, and we will henceforth call it a matching. The matching is easily obtained by the fol- lowing procedure:

Subroutine to construct the shortest matching:

Step I: For each of the 2m+ l points construct a candidate matching:

Step 1A: Start at the distinguished point and move clockwise around the carousel. Match the first-encountered point with the second-encountered point, the third with the fourth, etc. (The distin- guished point is not matched at all.)

Step 2: Choose the matching of shortest tool length from the 2m + 1 matchings from Step 1.

One difficulty remains. The matching of order end- points may fail to give an uninterrupted sequence in which t o pick the orders since the endpoints of orders may he paired in such a way as to yield disjoint circuits rather than a single path traceable by the picker. However we can easily correct this by connecting the disjoint circuits in a simple way. The entire procedure is formalized as follows:

"Hierarchical" Heuristic

Step 1: Construct the minimum spanning interval of each order.

Step 2: Construct the shortest matching among the 2m endpoints of orders and the start position o r the picker.

Step 3: If the current picker position is matched to an endpoint of an unpicked order, then travel to that endpoint and pick that order. Repeat Step 3.

Step 4: If any unpicked orders remain, travel clockwise until an unpicked endpoint is encountered, then execute Step 3.

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Figures 4, 5, 6 illustrate the steps of Hierarchical.

Figure 4: The minimum spanning intervals of a collection or orders. The items labeled icomorise order i li = 1 . . .51. For examole, order ~ ~

1 consistsof 3 items. he minimum spanning intervals connect the first and the last items nicked in an order ithouqh no direction is speci-

Figure 6. The sequence of picks determined by the Hierarchical heuristic. Order 3 is picked first as the carousel rotates clockwise. Then the carousel reverses direction and order 2 is picked. Then orders 1 .5 and 4 are picked. This is guaranteed to be no more than 1 carousel revolution longer than the shortest possible.

. - tied yet). To pick any brder requiresat least time proponional to the Notice that Hierarchical specifies a sequence of length of its minimum spanning interval. retrieval for the orders and, in addition, specifies a

sequence of retrieval for the items within each order. Thus it gives a complete solution to the two level sequencing problem.

THEOREM 3: Under Hierarchical, the carousel will never travel more than I revolution more than the short- est possible retrieval sequence.

NOTE: This guarantee is indeperzdent of the number of orders and the number of items to be retrieved! Conse- quently Hierarchical is asymptotically optimal (i.e. as the number of orders to be retrieved gets larger, the travel time required beyond optimal becomes negligible).

Proof: Let EI, be the sum of the lengths of the short- est spanning intervals of the orders, and let M be the length of the minimum-cost matching. Then any way of retrieving the orders requires total retrieval distance at least XI , + M. Hierarchical requires additional travel of at most one revolution of the carousel due to Step 3.

Hierarchical runs in O(Cn,) steps beyond the preprocessing sort, where n, is the number of items in order j . Thus, for little effort, we can determine a

F~aure 5 The dotted arcs are the minimum-cost maximal matchma orovablv near-o~timal seauence in which to retrieve a col- - . .~~ ~ ~ ~ - ~ , on the endpoints of the orders and the stan Position Of the picker. lection of orders. Moreover, as the number of orders At least this much additional travel (beyond the minimum spanning increases, the relative amount of unnecessary retrieval intervals) is required to pick the orders.

time Quickly becomes negligible. For sufficiently large item-density the following sim-

pler heuristic is appropriate.

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"Nearest Order" Heuristic

Step 1: Construct the minimum spanning interval of each order.

Step 2: Travel to the closest endpoint of an unpicked order and pick that order.

Nearest Order also requires O(Enj) steps, but it is much easier to program. Furthermore, the quality of its solutions can also be guaranteed: if the carousel has S storage locations, then

THEOREM 4: Under Nearest Order, the carousel will never travel more than log$ revolutions beyond optimum.

This guarantee is not as good as that for Hierarchical: nevertheless, it too is independent of the number of ord- ers and the number of items to be retrieved, and so is aymptotically optimal.

Proof: First note that once begun, each order is retrieved as quickly as is possible. Therefore any travel beyond optimum must he due to travel between orders. Call the travel from the end of one order to the begin- ning of the subsequent order a "gap". Gaps never partially overlap: Any gap is either not contained in any subsequent gaps, or else is contained entirely within a sub- sequent gap that is at least twice as long. This holds since otherwise the carousel must not have rotated to the nearest endpoint, in violation of the heuristic rule. Thus gaps are nested, with each successive gap at least twice the length of the preceding. The shortest gap must be at least 1/S revolutions since there are S storage locations; in addition no gap can be longer than 1/2 revolution. Thus there cannot be more than log,S gaps overlapping any point, and the carousel can have rotated past any point idle no more than log,S times. Furthermore, between subsequent rotations past any point the gaps sum to length no more than one carousel revolution. There- fore the total length of all gaps cannot exceed log,S revolutions.

The above argument in fact establishes a more general result: under the heuristic which always travels next to the order whose closest unpicked item is closest of all, the total distance travelled between orders is never greater than log, S, and this is independent of the sequences in which the items are retrieved within the orders. Thus using Nearest Order between orders and, say, Nearest Item wi- thin each order never requires travel beyond log, S + 2 * optimum, which for large numbers of orders is asymp- totically about twice optimum.

For still larger item-density, an even simpler heuristic is to be considered.

"Monomaniacal" Heuristic #2

Step 1: Always turn clockwise and pick the next encoun- tered order.

In the worst-case this greedy procedure can suggest

retrieval routes that are quite poor. Nevertheless, certain weak performance guarantees can be established 121. Despite the weak guarantees, when the item-density is sufficiently large, most carousel locations must be visit- ed, so that this simple heuristic sequences orders nearly as well as possible.

Conclusions

The appropriate choice of heuristic for retrieval de- pends on both the order-rate and the item-density. As these increase, we are justified in using simpler heuristics to sequence retrievals. These simpler heuristics may be capable of undesirable worst-case performance, but a large item-density will protect them from exhibiting pathological behavior. In this case they may be expected to produce retrieval routes that are negligibly longer than optimal routes.

Practical choice of appropriate retrieval algorithms should probably be done by simulation since it depends on (in addition to the item-density and order-rate) the probability of each item of being in a typical order, the locations in which those items are stored, the actual times required to pick items, and the speed at which the carousel rotates.

Informal tests suggest that, across a wide range of con- ditions Nearest item and Nearest Order are the algorithms of choice. An immediate advantage is that they continue to work as guaranteed even when each item may be stored in more than a single location. In general they perform close to the optimal for even fairly low item-densities; they produce solutions that are guaranteed to never be "too far" from optimal, independently of the number of items and orders they process. Furthermore, like others of the simple heuristics, they enjoy certain advantages, such as ease of implementation and speed of execution. This is especially useful in those situations where the carousel is under the control of a human picker, for whom it is important that these heuristics are simple to understand and can be implemented with no computa- tion. Finally, since these heuristics are "greedy" al- gorithms with little or no look-ahead, a tentative sequence of retrievals can be updated in real-time as the problem changes (as, for example, when more orders arrive, or items are cancelled from an order). Thus they can oper- ate in a dynamic environment in which the problem is changing even as it is being solved.

ACKNOWLEDGEMENTS

The authors thank R. C. Wilson for first interesting us in automated storage-and-retrieval, and H. D. Rat- liff and John A. White for support and helpful discus- sions. We also appreciate the cooperation of Bill Woo of White Data Systems, manufacturers of carousel con- veyors. Finally, we thank an anonymous referee whose suggestions improved the presentation of this paper.

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REFERENCES [I] A. V. Aho, J . E. Hopcroft, and I. D. Ullman (1974). The De-

sign and Analysis of Computer Algorithms, Addison-Wesley. [Z] J . J. Bartholdi and L. K. Platzman (1984). "Automated Retrieval

From Carousel Conveyors", Material Handling Center Report MHRC-TR-84-01, Georgia Institute of Technology.

131 E. L. Lawler, J . K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmays, eds. (1985). The Travelling Solesmon Problem, John Wiley.

Jahn Barlltold~ I I U J I ~ ut the Lnn~r.bl) of 1 lorlda, \\ hcrc hc cnrncd drptees in mathemau.. asJ Jpcr*llJnr rc*car.tt He ha, taught a1 lhc university of Michigan and the Shanghai Institute of Mechanical En- gineering. He is now Associate Professor in the School of Industrial and Systems Engineering at Georgia Institute of Technology, and mem- ber of the Production and Distribution Research Center and the Material Handling Research Center. In 1984 he was named a "Presidential Young Investigator" by the National Science Foundation.

Loren Platzman holds a doctorate in Electrical Engineering and Com- puter Science from MIT. He has been a member of the Technical Staff a t Bell Telephone Laboratories and, taught at the University of Michi- gan before joining the faculty of Georgia Institute of Technology, where he is currently Associate Professor of Industrial and Systems Engineer- ing. He is a co-founder of Technical Support Software. Inc., a firm specializing in customized decision support and automated control pro- grams for manufacturing and material handling systems.

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