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A perspective on location science
Locnrion Science, Vol. 5, No. 1, pp. 3-13, 1997 0 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0966-8349197 $17.00+0.Ml
PII:SO966-8349(97)00008-9
A PERSPECTIVE ON LOCATION SCIENCE CHARLES
REVELLE
Department of Geography and Environmental Engineering, The John Hopkins University, Baltimore, MD 21218-2686, USA
When the editors of Location Science asked me to write a perspective piece for the journal, I explored the variants of the word perspective. I recalled the adjective that means ‘being able to provide perspective’ or ‘having perspective’. That word is ‘perspicacious’. I once used that word, but just once. I was recommending a student to graduate school in public policy at Massachusetts Institute of Technology, and I referred to him as ‘perspicacious’. The chair of the admissions committee was an old friend, and he called me about the student. He told me that the committee had decided to accept him anyway. Thus, I will try to provide perspective without being perspicacious. Or, as I said to John Current when he asked for this piece by February, “No sweat”. Unfortunately, I did not deliver this until mid-April. The circumstance that warranted this invitation to provide a perspective was the Lifetime Achievement Award I received from the Section on Location Analysis (SOLA) of INFORMS in June, 1996. It is an honor to have received this award and to share it with Dick Francis. I am acutely aware, however, that Leon Cooper, had he lived, would undoubtedly have remained productive and would have been the odds-on choice to compete with Dick for the award. I am aware also that the portion of the award that went to me does not belong to me alone but to my location collaborators over the years, who are (in alphabetical order): Laurence Bergman Geoffrey Berlin David Bigman Jack Brimberg Richard Church Jared Cohon John Current Mark Daskin Thomas Eagles David Eaton Jack Elzinga David Engberg
Erhan Erkut Louis Falkson Roberto Galvao Miriam Heller Kathleen Hogan Vicki Hutson Gilbert Laporte Jon Liebman Vladimir Marianov David Marks George Moore Sam Ratick
Peter Rojeski Kenneth Rosing Hester Rosing-Vogelaar David Schilling Joseph Schweitzer Daniel Serra Don Shobrys Stephanie Snyder Ralph Swain Costas Toregas Earl Whitlatch Justin Williams Jeff Wright Zhong Ping Zhu
I hope 1 have not left anyone off this list who deserves to be there. At the outset, it is a privilege to be able to write these thoughts for an audience I respect and admire. Before I begin the piece formally, I want to offer a name for our field. While the name Location Science is an adequate and meaningful name, most sciences draw their names from 3
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Greek roots. The Greek for location is topothesia. So our field may be rightly called Topothesiology* and the journal subtitled. The Journal of Topothesiologv. The creation of the award represents to me a sign of the growing stature of location science. We are a time in the development of our field when we can look backward at where we have been and sideways at allied fields and forward. And we can assess our potential for growth and achievement as well as the directions we may go. I will try to cover all of the elements listed in this perspective piece. First, I think the productivity and advancement of our field is virtually assured. This statement is supported not only by the rich challenges that present themselves, the growing power of our computers, and our evolving mathematics. It is assured as well by the spirit of cooperative inquiry that pervades our discipline. I suggested in brief words at the time of the SOLA award that our field may nearly be a connected network. From any researcher, a path of collaboration may reach nearly every other researcher of our field. That is, A and B, though they may have never collaborated directly and may even be distant in interests, may be linked by association through a chain of cooperative researchers that include i, j, k, and 1. Ours is a unique field in its cooperation and it is this spirit that absolutely ensures our vitality. People in our field want to work together. I know why I work with colleagues because it makes research fun when we share ideas. Obviously, I regard research as an enjoyable activity rather than as work. Cooperative research is like a game of ‘leap frog’, in which each advance leads to another. Newton put it more eloquently, “If I have seen further than Descartes, it is by standing on the shoulders of giants.” My own pedestrian view would be that one pushes and another pulls the first over the fence. Whatever the motivation for this wonderful spirit of cooperative research, its presence is keenly felt and promises us decades of joyous inquiry. Who are we? Where does our tradition begin? Who was the first location scientist, in spirit at least? I will sketch a brief and idiosyncratic history, but the reader with a real taste for history should read Wesolowsky, 1993 or Rosing, 1991. Before location science could exist, distance had to exist, and while we instinctively understand the concept of distance on a line or on a network, most people credit Euclid and Pythagoras as the first to construct geometric models of distance. What a proud tradition, that of the ancient Greek geometers, in which to cast our roles. But the first person I am aware of to pose formally a location problem and suggest a solution was the Emperor Constantine (I am indebted to John Karkazis for this recognition. John’s research association, which has interests in location, environment and planning, is named the Constantine Porphyrogenetus Association). Interestingly, Constantine was solving a location problem on a network with only discrete positions available for the placement of the Roman legions. Six of these legions were called a ‘pebble’, and one can imagine why they were so named since the problem arose in the 4th century AD. The pebbles were deployed strategically so that with but a single troop movement each of the outposts of the far-flung empire could be defended against threats from invaders or from local insurgency; see Arquilla and Fredricksen, 1995. Thereafter, the lineage jumps to Fermat and Torricelli (both 17th century mathematicians) who posed the problem (with three nodes) that we call the Weber Problem today; i.e., the infinite solution space Euclidean distance minisum single site location problem, or to use George Wesolowsky’s nomination for its name, the Euclidean minisum problem, This *This name was suggested
to me by Charles
Branas,
a doctoral
student
currently
working with me.
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problem is also often called the Steiner-Weber problem because the single site problem on three nodes coincides with a problem of network design posed by Steiner. Little more of interest occurred until the 19th century when Sylvester, a mathematician, studied the infinite solution space minimax Euclidean distance problem. Sylvester was so famous when he came to a US university from England, that he could demand that he be paid in gold. Oh for the good old days of location research!; see Sylvester, 1857. The next location activity of note occurred in the 20th Century. In the original problem posed by Steiner, there were just three nodes, but in the general Steiner problem, multiple nodes are connected into a tree network by the least length of string. The network string need not only go from node to node as in the minimal spanning tree but can have junctures intermediate between nodes. These intermediate junctions are referred to as Steiner points. If one requires the least length network that connects a group of just three nodes at a single point, the Steiner problem and the Weber problem are identical. Thought of another way, suppose a Steiner tree has been constructed (not an easy task) on a number of nodes and three of the original nodes are connected to one another in a ‘y’ network where the intermediate point at the junction is new. The junction of the ‘y’ is also the solution to the Weber problem on those three nodes if the nodes are assumed to have equal weights. Thus, from the modem beginning of location research, the design of networks and siting have been intimately connected. Weber (1909) was an economist studying the location of industry, and the economist’s concern for the location of industry and commerce has continued to the present spurred by such people as Hotelling and Hoover. Hotelling (1929) created what later became the famous ‘ice cream vendor on a beach’, in which a new vendor attempts to capture the maximum market share from the first vendor who is located at the center of the beach. This is the problem that presages the area we collectively today refer to as ‘competitive location’. Hoover (1948) and Pallander (see Isard, 1956) pursued the single-factory Weber problem with multiple demands and multiple sources of supply, not just the three nodes of the original problem posed by Weber. They independently described the creation of lines of equal transport cost to supply each demand point around each raw material source. They then constructed lines of equal total transport cost which enclosed the unknown location of the plant, producing thereby contour lines, known as isodapanes, which described the total cost function for supplying the demands in concentric closed curves. They were thus pursuing the multi-source, multi-demand Weber problem. It was also in this era that Weiszfeld (1937) offered his iterative procedure for the Weber problem in an obscure journal. Although independently discovered about 1960 by Cooper (1963) Kuhn and Kuenne (1962) and Miehle (1958) that early solution method had virtually no effect on the field since it sank out of sight. It was not until 1958, with the computer revolution underway, that Baumol and Wolfe (1958) offered a mathematical programming formulation and approach for the warehouse location problem on a network. They were, as far as I know, the first to use the computer to ‘solve’ a location problem of any kind. Wersan et al. (1962) offered a formulation of the first rectilinear location problem, a minisum, single site location problem. These papers, even though the second is more obscure, were the first to see location in the mathematical programming/optimization framework. Hakimi’s classic proofs (1963, 1964) of nodal sufficiency for optimal siting in minisum problems on a network followed quickly. Our field was suddenly underway. I was at the time but a lad having just entered graduate school and
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wondering how I would spend my professional though I was totally unaware of them.
life. Dark forces were gathering
even then
1. WHO WE ARE TODAY What strikes me as unique to our mathematical field is the number of academic disciplines represented among us. There are electrical engineers, regional scientists, mechanical and industrial engineers, geographers, civil and environmental engineers, economists, computer scientists, operations researchers, chemical engineers, health policy analysts, regional planners, transportation scientists and engineers, and mathematical scientists. Can any other field lay claim to such a wide set of involved disciplines? Even though the first location problem I ever thought about was the Euclidean minisum problem, I drifted quickly to the ‘discrete’ camp of researchers who try to solve location problems on a network where only certain pre-specified sites are eligible to have facilities. I shifted to these problems because they were interesting practical problems and because I was too unsophisticated (read ‘wet behind the ears’) to realize that these zero-one programs were enormously difficult to solve exactly in every instance. I persist in working on these problems because my colleagues and I are not done making the problems more realistic. We continue to create new problem statements with steadily greater applicability. Twenty-five years later, we still are making covering problems capture more of reality. Theoretical challenges almost always accompany these new problem statements as well. Furthermore, the tools from solving one problem often help us to solve the next. Integer programming is, of course, the sine qua rzun for the solution of discrete solution space location problems. The last large step in integer programming was the invention of branch and bound by Land and Dolg (1960). This procedure, though initially a band-aid for situations with a few zero-one variables, has been automated and fine-tuned over the years because of advancing computational abilities - so that some location problems with hundreds, even thousands, of zero-one variables may now be attempted. To attack these discrete location problems requires either (1) an integer friendly formulation to which to apply linear programming and branch and bound or (2) heuristic procedures, By an integer friendly formulation, I mean a formulation for which branch and bound is often not needed at all or that only a limited amount of branch and bound is required when the add-on procedure is necessary (ReVelle, 1993). Progress in integer programming is slow and genuine breakthroughs have not occurred. The past three decades have, however, brought many successes in which particular zero-one programming problems have been effectively solved, especially location problems. Integer friendly formulations for the plant location problem, the p-median problem, the location set covering problem, the maximal covering problem and its variants have all been created. And Lagrangian relaxation has been applied to effectively solve many of these problems as well. It is especially interesting and relevant to recall that the fixed charge problem - of which the uncapacitated plant location problem is a prime example - is still an unsolved problem in the general case. Nonetheless, it can be considered solved in the specific case of plant location because of the integer friendliness of the Balinski-Morris formulation (Balinski, 1965; Morris, 1978). What is happening is that one-by-one, many problems that we would have regarded as unsolvable 15-20 years ago, are, for all practical purposes, solved now. Of course, computer scientists, concerned with solution in each and every case, might not think so, i.e. might not think that they are ‘solved’. But I am less interested in the search for
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perfection than the computer scientist. For me, life is too short, although I can appreciate that there is satisfaction to such a search. I would rather attempt to structure another problem and treat it to the same level of imperfect consideration than seek beauty and truth (an exact solution) in a single problem instance. Not that beauty and truth aren’t laudable goals, but so also is the discovery of new and useful problem statements. And I am convinced that many such new and useful problem statements are still to be created. Laporte and I (ReVelle and Laporte, 1996) described but did not solve a number of such problems, including simultaneous siting of plants and specialized machines within those plants. There exists also, in the process of formulating new models for new situations, the possibility of discovering new constraint forms that are integer friendly and that can be applied to problems of historical interest as well as to other new problems. As an example, in Snyder and ReVelle (1996a,b), we discovered that constraints denser than pairwise constraints, when applied to a certain problem of grid packing, produced zero-one solutions using only LP with extraordinary frequency. Subsequently, we were able to structure denser constraints than pairwise constraints for the r-separation problem (Erkut et al., 1996). These new denser constraints produced zero-one solutions with frequencies over 95% when linear programming alone was applied. In contrast, pairwise constraints almost always yielded solutions with all fractions. Daskin (1983) introduced a sequence of counting variables which individually were equal to one up to the sum which they counted. His model maximized the expected coverage of demands. Hogan and ReVelle (1986) also utilized counting variables in the context of calculating back up coverage of demands. We subsequently employed counting variables in structuring the maximum availability location problem (ReVelle and Hogan, 1989) and in models of availability in fire protection and ambulance service; see ReVelle and Marianov, 1991; Marianov and ReVelle, 1991. In Marianov and ReVelle (1992), we introduced an integer-friendly capacity constraint (p. 229) that is waiting for further exploitation. Another integer-friendly idea consists of the ingenious clipping constraints created for a network design problem by Church and Current (1993). These clipping constraints are so potent, so integer friendly, that they seem sure to find further application in other network design problems. Another integer-friendly idea is the adaptation/application of the shortest path formulation to problems which occur in time. We applied this idea to forest harvesting through time, where the goal is to maximize the value of a time stream of harvests (ReVelle and Snyder, 1996). Because the formulation is virtually the same as the shortest path formulation, integer solutions are almost assured. So far, since this formulation was created, I have found four additional industrial, environmental, and geographic application settings to which it can be applied, each one needing an integer friendly model for its statement/solution. None of these have yet been implemented. And I don’t doubt that others exist. Humbly, may I suggest you read this piece. If you beat me into print with a new application, at least it will be off my plate. The point I am trying to make is that the incremental approach to integer programming is working. We have no grand algorithm, but the problems are falling one-by-one, nonetheless, to new constraint forms or to heuristics or to a combination of these. My dissertation advisor and colleague, Walter Lynn, cautioned me once when I told him I was working on the general integer programming problem (I had one of those ideas that didn’t pan out). He said, “Chuck, don’t do it.” To this, I replied, “Why not?” He responded, “Integer programming is not really a form of mathematical programming.” I asked, “Then,
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what is it?” “It is”, Walter responded unhesitatingly, “a specialized form of insanity.” I think Walter is still right. I have had therapy and recovered, and the general zero-one programming algorithm still eludes us all. Now, I attack it obliquely rather than frontally, and I sleep better at night. To summarize this line of thought, I have a sense that our art, proceeding incrementally, one problem at a time, has the potential to uncover ideas that will be very useful to the general advancement of the field of integer (especially zero-one) programming. I said earlier that I would be looking to the sides as well as forward and backward. The discussion of integer programming is one side. Another is computational geometry. The computational geometers, obsessed with the museum problem, could relax the formulation that requires all the walls to be seen and ask, in the spirit of the maximal covering problem, what the guard positions might be that can see 90% of the wall length or 95%. The guards might also be interested in rapid response, so a median style objective or a maximum distance constraint could be added or both. Museum problems are rich with possibilities. We may also be visited more often by computational geometers who see our problems as grist for their mill. The fact that I am more interested in new problems than truth and beauty in older problems does not mean that I regard heuristics that push toward the optimum with any lack of interest. I see heuristics and metaheuristics as hope not simply for linear and non-linear combinatorial problems which can be stated in mathematical forms. I see them as the means to attack problems which cannot (yet) even be formulated as linear or non-linear mathematical programs. Even this latter class of problems needs solution approaches. An example of such a problem that was solved heuristically before a linear integer program had been written is the p-median problem. Two heuristics for the p-median had been created (by Teitz and Bart (1968) and Maranzana (1964)) before a mathematical programming formulation had ever been offered. Teitz and Bart created a vertex substitution procedure; Maranzana developed a cluster-improvement, cluster-modification procedure. Certain competitive location problems in which there is reaction by existing outlets to the new entrants fall in this category at the present time (Serra and ReVelle, 1994). That is, the problem can be stated, but no IP/LP problem statement yet exists. The last ten years have witnessed a fascinating developing in heuristics - the creation of super or overarching heuristics or controllers that modify the rules of the base heuristic to avoid the pitfalls we refer to as local optima. This development of controlling or metaheuristics promises us an even closer approach to the solution of the combinatorial problems with which we struggle. I do not think we are done creating metaheuristics. Further, just as we know we can combine base heuristics for a given problem, we may also be able to combine metaheuristics. My last look to the side is to a new development that provides great opportunities to location scientists. I am speaking of geographical information systems, or GIS, that have been developed throughout the 1990s. GIS’s colorful display of information can turn the heads of and capture decision makers. Geographical information systems use up-to-date computer technology to create intriguing displays of solutions that are outcomes of - you guessed it - heuristic and/or exact algorithms developed by location scientists. I see two possibilities for major growth in the field of location science, both related to the major strengths of location science, viz. the relevance of the models to the decision makers. The first is to make extensive use of GIS and its possibilities, i.e. “join ‘em”. In other words, do not just develop algorithms but make sure that they are put into GIS systems, as this is
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where decision makers will look and find them. The other possibility is to push even more vigorously the front of applying multiobjective programming. Over the years, I have witnessed a number of occasions when the ability to elicit and incorporate multiple objectives in location problems made a difference in the level of acceptance and involvement by decision makers. In a sense, I see multiobjective tradeoff curves (such as population coverage vs. property coverage, for instance) as another hook that catches the decision maker. The GIS output may be used regularly for the display of near-optimal solutions, but so far I have not heard of its use for displaying solutions on the tradeoff curve between objectives. GIS could, of course, display solutions from a tradeoff curve, even though the exact curve requires the use of exact methods. I see such curves as value-added to location problems. They move the decision-makers to a higher level, to a deeper and more thoughtful consideration of alternatives than GIS alone could possibly provide. One open question is how to graphically display tradeoff curves in the case of more than three objectives. In another disciplinary area in which I participate, water resources, I see similar events occurring. Here, an old and less scientific technique, simulation, that, over the decades, was made simple to use by user-friendly interfaces, competes with optimization. And again, similar to my discussion above, optimization could take the same role as location science models, whereas simulation could be used as the same marketing or display tool as GIS. Our problem is who we are: We are researchers, most of us, interested in challenging location and network design problems. Few of us would turn down a consulting contract with Home Depot but who of us would seek such a contract? I have no answer for our predilection for thought as opposed to action.
2. NEW APPLICATIONS New applications. New methods. The two can proceed separately but they can also go handin-hand. Linear programming itself was not created as an abstract mathematics but as a means to solve such problems as resource allocation, goods distribution, and diet creation. Calculus was developed by Newton to solve problems of planetary motion. And I learned mathematics to solve problems as opposed to make me a creator of new mathematics. To the extent that I do work on new mathematics, it has always been in the context of a problem. Where are the new problems? This is the only question I am willing to approach since I don’t actively seek to create new mathematics as a primary activity. That is, I begin with a problem, and once I have it in words, have the objective and the constraints, I try to formulate it. If I can’t put the problem in integer-friendly form, I look to reformulate it or replace the offending constraints with less offensive, hopefully integer friendly ones. And if I can’t do that, I will move to heuristics. Is any area so mature that nothing is left to do? Here I will focus on discrete location problems and say that no areas seem fully mature. No area seems to be so complete that new and better problem statements as well as approaches are not possible. I will describe some of the areas I see as open. Laporte and I (1996) discussed in some detail some of the open areas of plant location. 1 will mention two of the problems from that article which I see as most important and three additional areas as well. First, there is the issue of siting simultaneously both plants and specialized machines that will be placed within those plants. Each specialized machine is
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characterized by specific product capabilities peculiar to that machine. Production costs are also machine specific. The machines may be capacitated giving rise to a staircase cost function as described in Holmborg (1994) or uncapacitated. Second, our model of individual shipments, plant to demand is not always correct. Often the requirements of multiple markets may be placed on a single truck for distribution via a tour. Models of this situation are now being formulated and solved (Berger et al., 1997). Areas of interest not mentioned in the ReVelle-Laporte article include (1) capacitated plant location, (2) the cost function of the plant, and (3) integrated plant location. In the past dozen years, investigators have focused more on capacitated plant location, often using Lagrangean relaxation. Beginning with the reports of success by Davis and Ray (1969), we have had conflicting messages. Often the messages are anecdotal. Davis and Ray claimed their linear-programming-equivalent methodology produced no fractional variables, whereas others found many fractions. It appears that the relation of the sum of capacities to sum of demands has a significant effect. Heller et al. (1989) and Rosing (1981) both find that as the sum of capacities approach within 3O%_t of the sum of demands, fractional facility variables increase in frequency. This little noticed phenomenon means that the successes reported by some researchers may be data-set dependent. New work, exploring the full range of ratios of capacity to demand, is needed. The concept of Heuristic Concentration (Rosing and ReVelle, 1997) could have a role in resolving some of these problems by reducing the solution space via heuristics, and then applying an exact method. The cost function of the plant is strongly related to the issue of capacity. A capacitated plant may be interpreted as a plant for which the cost of increasing production beyond some level becomes excessive. That is, a convex portion of the cost curve makes further production uneconomical. Thus, if after the fixed charge portion of the cost curve, a convex cost function can be successfully modeled, the issue of capacity could become irrelevant - in cases where capacity really means excessive cost of expansion. Finally, on the plant location side, integrated plant location is little explored. By integrated’ I mean that the components are layered - from the resource nodes to the plant nodes to the warehouse level to the final customer level. The papers which cover all the elements of this area can be counted on one hand. The area of network design or extensive facility location (Mesa and Boffey, 1996) is rich in applications. These include the siting or improvement of highway systems, the location or expansion of rail and transit systems, the design of airline networks, the determination of bus routes, the planning of collection and distribution routes, the construction of power line networks, and of telecommunication networks. Many of these networks have associated point facilities such as stops, hubs, garages, depots, switching stations, computers, concentrators, and the like. There is a richness of applications here. Issues within these areas which are in need of examination include the reliability of networks and the design of tree networks. Reliability of a system may not even be clearly defined mathematically, but reliability of service to a particular node does suggest multiple routes between the source node(s) and the demand node. As a consequence, the network designed to achieved some reliability standard is unlikely to be a tree. And even if a network designer had a problem statement in which a tree would do, it is not clear whether the tree ought to be a subtree of the minimal spanning tree, or any subtree which satisfies the constraints and optimizes the objective. For now, most applications I know of argue for and develop a subtree of the minimal spanning tree. It is easy to argue for the subtree of the MST, among other reasons because the more relaxed problem is unsolved, to my knowledge.
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Currently, I am, among other areas, interested (with my current and former students and collegues) in two related areas: (1) in the spatial and temporal management of forests for timber and wildlife purposes, and (2) in natural area planning in which parcels are assembled or connected into systems for the protection of multiple species. (See, for instance, Church et al., 1996; Snyder and ReVelle, 1996a,b; Snyder and ReVelle, 1997; Williams and ReVelle, 1996, 1997.) The combinatorial aspects of location problems are displayed here and the further challenge of temporal decisions and spatial compactness of assemblages make the problems fascinating. The need for networks of assemblages and road networks in forest management make the problems more interesting yet. I am also interested in emergency services which combine both ground and air transportation for victims of trauma. These problems are complicated by the presence of interacting services and facilities - helicopters and trauma centers. Finally, I think there is opportunity in two related problem settings - the clustering problem and the quadratic assignment problem. The clustering problem seeks the ‘natural clusters’ of data or areas. The goodness of the clustering involves either the nearness of members of the clusters to each other or the nearness of members to some central member of the cluster or the tightness or area of the polygon enclosing the members in each cluster. Obviously, the second of these is a p-median style problem. The first of these is akin to the objective of the quadratic assignment problem (QAP) in that the nearness of two nodes is only counted if the two nodes are in the same cluster - just as in the QAP where the interaction is weighted by the distance between the activities, which is determined by their placement in the grid. The QAP is both a problem from industry and a problem in the spatial planning of metropolitan regions. My hunch is that the formulation or heuristic that solves one (clustering or the QAP) will also, suitably modified, solve the other. How do we foster the growth of our discipline? The simple answer, I think, is by being relevant, by attacking real problems. But we could use, in addition, a means to facilitate our research. I would like to suggest that all members of SOLA or readers of this piece who are US citizens write to Dr. Neal F. Lane, Director of the US National Science Foundation. Urge Dr. Lane to support the funding of a multi-university consortium center for location scholarship - a Center for Siting and Distribution Research in which scholars from industry and academia interact. Your letter should stress not only the importance of the research to the competitiveness of US industry and commerce - with as much detail as possible - but also the importance of our research to governmental institutions and the public sector, including the military, the post office, hospital and trauma systems, ambulance and fire systems, national forest management, regional management of water pollution, solid waste management, school systems, and species preservation through land set aside. Urge that the center should have wide membership and should bring scholars from overseas to interact with American scholars. Suggest also that the center have an advisory board that reflects society’s locational needs including industrialists, heads of store chains, hospital systems administrators, school system administrators, public works directors and the like. The address is Dr. N.F. Lane, Director, US National Science Foundation, 4201 Wilson Boulevard, Arlington, VA 22230. Even if such a center is never created, we will continue to be vital and productive because we offer alternatives to real problems that people face daily. People make siting decisions every day, important decisions that influence the profits of industry, life-and-death decisions for emergency services, bottom-line decisions for retail firms, and strategic decisions for
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defense. With an agenda like this, and with a dedicated our success is assured.
and cooperating
core of researchers,
REFERENCES H. (1995) Graphing an optimal grand strategy. Milituy Opertttions Research, Fall, 3-17. Balinski, M. (1965) Integer Programming: Methods, uses, computations. Management Science, 12,253-311. Baumol, W. & Wolfe, P. (1958) A warehouse-location problem. Operations Reseurch, 6, 252-263. Berger, R., Coullard, C. & Daskin, M. (1997) Modelling and Solving Location - Routing Problems, Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL. Church, R. & Current, J. (1993) Maximal covering tree problems. Nuval Research Logistics, 4, 129-142. Church, R.. Stom, D. & Davis, F. (1996) Reserve selection as a maximal covering location problem. Biological Conservutron, 76, 105-112. Cooper, L. (1963) Location-allocation problems. Operations Research, 11, 331-343. Daskin, M. (1983) A maximum expected covering location model formulation, properties and heuristic solution. Tkznsponation Science, 17,48-70. Davis and Ray, 1969. Erkut, E., ReVelle, C. & Ulktisal, Y. (1996) Integer-friendly formulations for the r-separation problem. European Journal of Operational Research, 92, 342-35 1. Heller, M., Cohon, J. & ReVelle, C. (1989) The use of simulation in validating a multi-objective EMS location model. Annuls of Operations Research, 18, 303-322. Hogan, K. & ReVelle, C. (1986) Concepts and applications of backup coverage. Management Science, 32, 1434-1444. Holmborg, K. (1994) Solving the staircase cost-facility location problem with decomposition and piecewise linearization. European Journal of Operational Research, 75,41-61. Hoover, E. (1948) The Location of Economic A&@. New York. Hotelling, H. (1929) Stability in competition. Economic Journal, 39, 41-57. Isard, W. (1956) Location and Space Economy. Technology Press of MIT and Wiley, New York. Kuhn, H. & Kuenne, R. (1962) An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. Journal of Regional Science, 4, 21-34. Land, A. & Doig, A. (1960) An automatic method of solving discrete programming problems. Economettictt, 28, 497-520. Osman, I. & Kelly, J. (1995) M&-Heuristics: Theoy and Applications. Kluwer, Dordrecht, 407 pp. Maranzana, F. (1964) On the location of supply points to minimize transport costs. Operations Research Quarterly, 15,261. Marianov, V. & ReVelle, C. (1991) The standard response fire protection siting model. INFOR, 29, 116-129. Marianov, V. & ReVelle, C. (1992) A probabilistic fire protection siting model with joint vehicle reliability Papers in Regional Science, 71, 217-241. requirements. Mesa, J. & Boffey, T. B. (1996) A review of extensive facility location in networks. European Journal of Operutional Research, 95, 592-603. Miehle, W. (1958) Link-length minimization in networks. Operations Research, 6, 232-243. Morris, J. (1978) On the extent to which certain fixed charge depot location problems can be solved by LP. Journal of the Operational Research Society, 29, 71-76. ReVelle, C. & Hogan, K. (1989) The Maximum availability location problem. Trunspontttion Science, 23, 192-202. ReVelle, C. & Laporte, G. (1996) The Plant Location Problem: New Models and Research Prospects. Operations Research, 44, p. 864. ReVelle, C. & Marianov, V. (1991) A probabilistic FLEET model with individual vehicle reliability requirements. European Journal of Operationul Research, 53, 93-105. ReVelle, C. (1993) Facility siting and integer friendly programming. European Journal of Operational Reseurch, 65, 147-158. ReVelle, C. & Snyder, S. (1996) A shortest path model for the optimal timing of forest harvest decisions. Environment and Planning, B 23, 165- 175. Rosing, K. (1981) Personal Communication, June. Rosing, K. (1991) Towards the solution of the (generalized) multi-Weber problem. Environment and Planning R, Planning and Design, 18, 347-360. Rosing, K. & ReVelle, C. (1997) Heuristic concentration: Two stage solution construction. European Journal of Operational Research, 97, 75.
Arquilla, J. & Fredricksen,
A perspective
on location
I3
science
Serra, D. & ReVelle, C. (1994) Market capture by two competitors: The pre-emptive location problem. Journal of Regional Science, 34, 549-56 I, Snyder, S. & ReVelle, C. (1996) The grid packing problem. Forevt Science. 42, 27-34. Snyder, S. & ReVelle, C. (1997) Dynamic selection of harvests with adjacency restrictions: the SHARE model. Forest Science.
Snyder,
S. & ReVelle,
of Forest Research,
C. (1996) Temporal
and spatial
harvesting
of irregular
parcel
systems. C’anudian
Sylvester, J. (1857) A question in the geometry of situation. nuurferly Journal of’Mathemutlc~s, 1, 79. Teitz, M. & Bart, P. (1968) Heuristic methods for estimating the generalized vertex median of a weighted 0prrution.s
Journal
26, 1079- 1088.
graph.
Research, 16, 955.
Weher, 19.57. Uber den Standort der Industrien (On Location of Industries) 1909 (English translation by C. Friedrich (1957)). University of Chicago Press, Chicago. Weiszfeld. E. V. (1937) Sur le point lequal la somme des distances de n points don&s est minimum. T/te Tohoku Muthematical
Journal,
43, 335-386.
Wersan, S., Quon, J. & Charnes, A. (1962) American Public, Works Kwrbook. Wesolowsky. G. (1993) The Weber problem: History and perspectives. Location Science, 1, S-23. Williams. J. & ReVelle, C. (1996) A O-l programming approach to delineating protected reserves. Environmmr and Planning, B 23, 607-624. Williams, J. & ReVelle, C. (1997) Reserve assemblage of critical areas: A O-I programming approach. European Jownul of Operulional Research (to appear),
PII:SO966-8349(97)00008-9
A PERSPECTIVE ON LOCATION SCIENCE CHARLES
REVELLE
Department of Geography and Environmental Engineering, The John Hopkins University, Baltimore, MD 21218-2686, USA
When the editors of Location Science asked me to write a perspective piece for the journal, I explored the variants of the word perspective. I recalled the adjective that means ‘being able to provide perspective’ or ‘having perspective’. That word is ‘perspicacious’. I once used that word, but just once. I was recommending a student to graduate school in public policy at Massachusetts Institute of Technology, and I referred to him as ‘perspicacious’. The chair of the admissions committee was an old friend, and he called me about the student. He told me that the committee had decided to accept him anyway. Thus, I will try to provide perspective without being perspicacious. Or, as I said to John Current when he asked for this piece by February, “No sweat”. Unfortunately, I did not deliver this until mid-April. The circumstance that warranted this invitation to provide a perspective was the Lifetime Achievement Award I received from the Section on Location Analysis (SOLA) of INFORMS in June, 1996. It is an honor to have received this award and to share it with Dick Francis. I am acutely aware, however, that Leon Cooper, had he lived, would undoubtedly have remained productive and would have been the odds-on choice to compete with Dick for the award. I am aware also that the portion of the award that went to me does not belong to me alone but to my location collaborators over the years, who are (in alphabetical order): Laurence Bergman Geoffrey Berlin David Bigman Jack Brimberg Richard Church Jared Cohon John Current Mark Daskin Thomas Eagles David Eaton Jack Elzinga David Engberg
Erhan Erkut Louis Falkson Roberto Galvao Miriam Heller Kathleen Hogan Vicki Hutson Gilbert Laporte Jon Liebman Vladimir Marianov David Marks George Moore Sam Ratick
Peter Rojeski Kenneth Rosing Hester Rosing-Vogelaar David Schilling Joseph Schweitzer Daniel Serra Don Shobrys Stephanie Snyder Ralph Swain Costas Toregas Earl Whitlatch Justin Williams Jeff Wright Zhong Ping Zhu
I hope 1 have not left anyone off this list who deserves to be there. At the outset, it is a privilege to be able to write these thoughts for an audience I respect and admire. Before I begin the piece formally, I want to offer a name for our field. While the name Location Science is an adequate and meaningful name, most sciences draw their names from 3
4
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REVELLE
Greek roots. The Greek for location is topothesia. So our field may be rightly called Topothesiology* and the journal subtitled. The Journal of Topothesiologv. The creation of the award represents to me a sign of the growing stature of location science. We are a time in the development of our field when we can look backward at where we have been and sideways at allied fields and forward. And we can assess our potential for growth and achievement as well as the directions we may go. I will try to cover all of the elements listed in this perspective piece. First, I think the productivity and advancement of our field is virtually assured. This statement is supported not only by the rich challenges that present themselves, the growing power of our computers, and our evolving mathematics. It is assured as well by the spirit of cooperative inquiry that pervades our discipline. I suggested in brief words at the time of the SOLA award that our field may nearly be a connected network. From any researcher, a path of collaboration may reach nearly every other researcher of our field. That is, A and B, though they may have never collaborated directly and may even be distant in interests, may be linked by association through a chain of cooperative researchers that include i, j, k, and 1. Ours is a unique field in its cooperation and it is this spirit that absolutely ensures our vitality. People in our field want to work together. I know why I work with colleagues because it makes research fun when we share ideas. Obviously, I regard research as an enjoyable activity rather than as work. Cooperative research is like a game of ‘leap frog’, in which each advance leads to another. Newton put it more eloquently, “If I have seen further than Descartes, it is by standing on the shoulders of giants.” My own pedestrian view would be that one pushes and another pulls the first over the fence. Whatever the motivation for this wonderful spirit of cooperative research, its presence is keenly felt and promises us decades of joyous inquiry. Who are we? Where does our tradition begin? Who was the first location scientist, in spirit at least? I will sketch a brief and idiosyncratic history, but the reader with a real taste for history should read Wesolowsky, 1993 or Rosing, 1991. Before location science could exist, distance had to exist, and while we instinctively understand the concept of distance on a line or on a network, most people credit Euclid and Pythagoras as the first to construct geometric models of distance. What a proud tradition, that of the ancient Greek geometers, in which to cast our roles. But the first person I am aware of to pose formally a location problem and suggest a solution was the Emperor Constantine (I am indebted to John Karkazis for this recognition. John’s research association, which has interests in location, environment and planning, is named the Constantine Porphyrogenetus Association). Interestingly, Constantine was solving a location problem on a network with only discrete positions available for the placement of the Roman legions. Six of these legions were called a ‘pebble’, and one can imagine why they were so named since the problem arose in the 4th century AD. The pebbles were deployed strategically so that with but a single troop movement each of the outposts of the far-flung empire could be defended against threats from invaders or from local insurgency; see Arquilla and Fredricksen, 1995. Thereafter, the lineage jumps to Fermat and Torricelli (both 17th century mathematicians) who posed the problem (with three nodes) that we call the Weber Problem today; i.e., the infinite solution space Euclidean distance minisum single site location problem, or to use George Wesolowsky’s nomination for its name, the Euclidean minisum problem, This *This name was suggested
to me by Charles
Branas,
a doctoral
student
currently
working with me.
A perspective
on location
science
5
problem is also often called the Steiner-Weber problem because the single site problem on three nodes coincides with a problem of network design posed by Steiner. Little more of interest occurred until the 19th century when Sylvester, a mathematician, studied the infinite solution space minimax Euclidean distance problem. Sylvester was so famous when he came to a US university from England, that he could demand that he be paid in gold. Oh for the good old days of location research!; see Sylvester, 1857. The next location activity of note occurred in the 20th Century. In the original problem posed by Steiner, there were just three nodes, but in the general Steiner problem, multiple nodes are connected into a tree network by the least length of string. The network string need not only go from node to node as in the minimal spanning tree but can have junctures intermediate between nodes. These intermediate junctions are referred to as Steiner points. If one requires the least length network that connects a group of just three nodes at a single point, the Steiner problem and the Weber problem are identical. Thought of another way, suppose a Steiner tree has been constructed (not an easy task) on a number of nodes and three of the original nodes are connected to one another in a ‘y’ network where the intermediate point at the junction is new. The junction of the ‘y’ is also the solution to the Weber problem on those three nodes if the nodes are assumed to have equal weights. Thus, from the modem beginning of location research, the design of networks and siting have been intimately connected. Weber (1909) was an economist studying the location of industry, and the economist’s concern for the location of industry and commerce has continued to the present spurred by such people as Hotelling and Hoover. Hotelling (1929) created what later became the famous ‘ice cream vendor on a beach’, in which a new vendor attempts to capture the maximum market share from the first vendor who is located at the center of the beach. This is the problem that presages the area we collectively today refer to as ‘competitive location’. Hoover (1948) and Pallander (see Isard, 1956) pursued the single-factory Weber problem with multiple demands and multiple sources of supply, not just the three nodes of the original problem posed by Weber. They independently described the creation of lines of equal transport cost to supply each demand point around each raw material source. They then constructed lines of equal total transport cost which enclosed the unknown location of the plant, producing thereby contour lines, known as isodapanes, which described the total cost function for supplying the demands in concentric closed curves. They were thus pursuing the multi-source, multi-demand Weber problem. It was also in this era that Weiszfeld (1937) offered his iterative procedure for the Weber problem in an obscure journal. Although independently discovered about 1960 by Cooper (1963) Kuhn and Kuenne (1962) and Miehle (1958) that early solution method had virtually no effect on the field since it sank out of sight. It was not until 1958, with the computer revolution underway, that Baumol and Wolfe (1958) offered a mathematical programming formulation and approach for the warehouse location problem on a network. They were, as far as I know, the first to use the computer to ‘solve’ a location problem of any kind. Wersan et al. (1962) offered a formulation of the first rectilinear location problem, a minisum, single site location problem. These papers, even though the second is more obscure, were the first to see location in the mathematical programming/optimization framework. Hakimi’s classic proofs (1963, 1964) of nodal sufficiency for optimal siting in minisum problems on a network followed quickly. Our field was suddenly underway. I was at the time but a lad having just entered graduate school and
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CHARLES REVELLE
wondering how I would spend my professional though I was totally unaware of them.
life. Dark forces were gathering
even then
1. WHO WE ARE TODAY What strikes me as unique to our mathematical field is the number of academic disciplines represented among us. There are electrical engineers, regional scientists, mechanical and industrial engineers, geographers, civil and environmental engineers, economists, computer scientists, operations researchers, chemical engineers, health policy analysts, regional planners, transportation scientists and engineers, and mathematical scientists. Can any other field lay claim to such a wide set of involved disciplines? Even though the first location problem I ever thought about was the Euclidean minisum problem, I drifted quickly to the ‘discrete’ camp of researchers who try to solve location problems on a network where only certain pre-specified sites are eligible to have facilities. I shifted to these problems because they were interesting practical problems and because I was too unsophisticated (read ‘wet behind the ears’) to realize that these zero-one programs were enormously difficult to solve exactly in every instance. I persist in working on these problems because my colleagues and I are not done making the problems more realistic. We continue to create new problem statements with steadily greater applicability. Twenty-five years later, we still are making covering problems capture more of reality. Theoretical challenges almost always accompany these new problem statements as well. Furthermore, the tools from solving one problem often help us to solve the next. Integer programming is, of course, the sine qua rzun for the solution of discrete solution space location problems. The last large step in integer programming was the invention of branch and bound by Land and Dolg (1960). This procedure, though initially a band-aid for situations with a few zero-one variables, has been automated and fine-tuned over the years because of advancing computational abilities - so that some location problems with hundreds, even thousands, of zero-one variables may now be attempted. To attack these discrete location problems requires either (1) an integer friendly formulation to which to apply linear programming and branch and bound or (2) heuristic procedures, By an integer friendly formulation, I mean a formulation for which branch and bound is often not needed at all or that only a limited amount of branch and bound is required when the add-on procedure is necessary (ReVelle, 1993). Progress in integer programming is slow and genuine breakthroughs have not occurred. The past three decades have, however, brought many successes in which particular zero-one programming problems have been effectively solved, especially location problems. Integer friendly formulations for the plant location problem, the p-median problem, the location set covering problem, the maximal covering problem and its variants have all been created. And Lagrangian relaxation has been applied to effectively solve many of these problems as well. It is especially interesting and relevant to recall that the fixed charge problem - of which the uncapacitated plant location problem is a prime example - is still an unsolved problem in the general case. Nonetheless, it can be considered solved in the specific case of plant location because of the integer friendliness of the Balinski-Morris formulation (Balinski, 1965; Morris, 1978). What is happening is that one-by-one, many problems that we would have regarded as unsolvable 15-20 years ago, are, for all practical purposes, solved now. Of course, computer scientists, concerned with solution in each and every case, might not think so, i.e. might not think that they are ‘solved’. But I am less interested in the search for
A perspective on location science
7
perfection than the computer scientist. For me, life is too short, although I can appreciate that there is satisfaction to such a search. I would rather attempt to structure another problem and treat it to the same level of imperfect consideration than seek beauty and truth (an exact solution) in a single problem instance. Not that beauty and truth aren’t laudable goals, but so also is the discovery of new and useful problem statements. And I am convinced that many such new and useful problem statements are still to be created. Laporte and I (ReVelle and Laporte, 1996) described but did not solve a number of such problems, including simultaneous siting of plants and specialized machines within those plants. There exists also, in the process of formulating new models for new situations, the possibility of discovering new constraint forms that are integer friendly and that can be applied to problems of historical interest as well as to other new problems. As an example, in Snyder and ReVelle (1996a,b), we discovered that constraints denser than pairwise constraints, when applied to a certain problem of grid packing, produced zero-one solutions using only LP with extraordinary frequency. Subsequently, we were able to structure denser constraints than pairwise constraints for the r-separation problem (Erkut et al., 1996). These new denser constraints produced zero-one solutions with frequencies over 95% when linear programming alone was applied. In contrast, pairwise constraints almost always yielded solutions with all fractions. Daskin (1983) introduced a sequence of counting variables which individually were equal to one up to the sum which they counted. His model maximized the expected coverage of demands. Hogan and ReVelle (1986) also utilized counting variables in the context of calculating back up coverage of demands. We subsequently employed counting variables in structuring the maximum availability location problem (ReVelle and Hogan, 1989) and in models of availability in fire protection and ambulance service; see ReVelle and Marianov, 1991; Marianov and ReVelle, 1991. In Marianov and ReVelle (1992), we introduced an integer-friendly capacity constraint (p. 229) that is waiting for further exploitation. Another integer-friendly idea consists of the ingenious clipping constraints created for a network design problem by Church and Current (1993). These clipping constraints are so potent, so integer friendly, that they seem sure to find further application in other network design problems. Another integer-friendly idea is the adaptation/application of the shortest path formulation to problems which occur in time. We applied this idea to forest harvesting through time, where the goal is to maximize the value of a time stream of harvests (ReVelle and Snyder, 1996). Because the formulation is virtually the same as the shortest path formulation, integer solutions are almost assured. So far, since this formulation was created, I have found four additional industrial, environmental, and geographic application settings to which it can be applied, each one needing an integer friendly model for its statement/solution. None of these have yet been implemented. And I don’t doubt that others exist. Humbly, may I suggest you read this piece. If you beat me into print with a new application, at least it will be off my plate. The point I am trying to make is that the incremental approach to integer programming is working. We have no grand algorithm, but the problems are falling one-by-one, nonetheless, to new constraint forms or to heuristics or to a combination of these. My dissertation advisor and colleague, Walter Lynn, cautioned me once when I told him I was working on the general integer programming problem (I had one of those ideas that didn’t pan out). He said, “Chuck, don’t do it.” To this, I replied, “Why not?” He responded, “Integer programming is not really a form of mathematical programming.” I asked, “Then,
8
CHARLES REVELLE
what is it?” “It is”, Walter responded unhesitatingly, “a specialized form of insanity.” I think Walter is still right. I have had therapy and recovered, and the general zero-one programming algorithm still eludes us all. Now, I attack it obliquely rather than frontally, and I sleep better at night. To summarize this line of thought, I have a sense that our art, proceeding incrementally, one problem at a time, has the potential to uncover ideas that will be very useful to the general advancement of the field of integer (especially zero-one) programming. I said earlier that I would be looking to the sides as well as forward and backward. The discussion of integer programming is one side. Another is computational geometry. The computational geometers, obsessed with the museum problem, could relax the formulation that requires all the walls to be seen and ask, in the spirit of the maximal covering problem, what the guard positions might be that can see 90% of the wall length or 95%. The guards might also be interested in rapid response, so a median style objective or a maximum distance constraint could be added or both. Museum problems are rich with possibilities. We may also be visited more often by computational geometers who see our problems as grist for their mill. The fact that I am more interested in new problems than truth and beauty in older problems does not mean that I regard heuristics that push toward the optimum with any lack of interest. I see heuristics and metaheuristics as hope not simply for linear and non-linear combinatorial problems which can be stated in mathematical forms. I see them as the means to attack problems which cannot (yet) even be formulated as linear or non-linear mathematical programs. Even this latter class of problems needs solution approaches. An example of such a problem that was solved heuristically before a linear integer program had been written is the p-median problem. Two heuristics for the p-median had been created (by Teitz and Bart (1968) and Maranzana (1964)) before a mathematical programming formulation had ever been offered. Teitz and Bart created a vertex substitution procedure; Maranzana developed a cluster-improvement, cluster-modification procedure. Certain competitive location problems in which there is reaction by existing outlets to the new entrants fall in this category at the present time (Serra and ReVelle, 1994). That is, the problem can be stated, but no IP/LP problem statement yet exists. The last ten years have witnessed a fascinating developing in heuristics - the creation of super or overarching heuristics or controllers that modify the rules of the base heuristic to avoid the pitfalls we refer to as local optima. This development of controlling or metaheuristics promises us an even closer approach to the solution of the combinatorial problems with which we struggle. I do not think we are done creating metaheuristics. Further, just as we know we can combine base heuristics for a given problem, we may also be able to combine metaheuristics. My last look to the side is to a new development that provides great opportunities to location scientists. I am speaking of geographical information systems, or GIS, that have been developed throughout the 1990s. GIS’s colorful display of information can turn the heads of and capture decision makers. Geographical information systems use up-to-date computer technology to create intriguing displays of solutions that are outcomes of - you guessed it - heuristic and/or exact algorithms developed by location scientists. I see two possibilities for major growth in the field of location science, both related to the major strengths of location science, viz. the relevance of the models to the decision makers. The first is to make extensive use of GIS and its possibilities, i.e. “join ‘em”. In other words, do not just develop algorithms but make sure that they are put into GIS systems, as this is
A perspective
on location
science
9
where decision makers will look and find them. The other possibility is to push even more vigorously the front of applying multiobjective programming. Over the years, I have witnessed a number of occasions when the ability to elicit and incorporate multiple objectives in location problems made a difference in the level of acceptance and involvement by decision makers. In a sense, I see multiobjective tradeoff curves (such as population coverage vs. property coverage, for instance) as another hook that catches the decision maker. The GIS output may be used regularly for the display of near-optimal solutions, but so far I have not heard of its use for displaying solutions on the tradeoff curve between objectives. GIS could, of course, display solutions from a tradeoff curve, even though the exact curve requires the use of exact methods. I see such curves as value-added to location problems. They move the decision-makers to a higher level, to a deeper and more thoughtful consideration of alternatives than GIS alone could possibly provide. One open question is how to graphically display tradeoff curves in the case of more than three objectives. In another disciplinary area in which I participate, water resources, I see similar events occurring. Here, an old and less scientific technique, simulation, that, over the decades, was made simple to use by user-friendly interfaces, competes with optimization. And again, similar to my discussion above, optimization could take the same role as location science models, whereas simulation could be used as the same marketing or display tool as GIS. Our problem is who we are: We are researchers, most of us, interested in challenging location and network design problems. Few of us would turn down a consulting contract with Home Depot but who of us would seek such a contract? I have no answer for our predilection for thought as opposed to action.
2. NEW APPLICATIONS New applications. New methods. The two can proceed separately but they can also go handin-hand. Linear programming itself was not created as an abstract mathematics but as a means to solve such problems as resource allocation, goods distribution, and diet creation. Calculus was developed by Newton to solve problems of planetary motion. And I learned mathematics to solve problems as opposed to make me a creator of new mathematics. To the extent that I do work on new mathematics, it has always been in the context of a problem. Where are the new problems? This is the only question I am willing to approach since I don’t actively seek to create new mathematics as a primary activity. That is, I begin with a problem, and once I have it in words, have the objective and the constraints, I try to formulate it. If I can’t put the problem in integer-friendly form, I look to reformulate it or replace the offending constraints with less offensive, hopefully integer friendly ones. And if I can’t do that, I will move to heuristics. Is any area so mature that nothing is left to do? Here I will focus on discrete location problems and say that no areas seem fully mature. No area seems to be so complete that new and better problem statements as well as approaches are not possible. I will describe some of the areas I see as open. Laporte and I (1996) discussed in some detail some of the open areas of plant location. 1 will mention two of the problems from that article which I see as most important and three additional areas as well. First, there is the issue of siting simultaneously both plants and specialized machines that will be placed within those plants. Each specialized machine is
10
CHARLES
REVELLE
characterized by specific product capabilities peculiar to that machine. Production costs are also machine specific. The machines may be capacitated giving rise to a staircase cost function as described in Holmborg (1994) or uncapacitated. Second, our model of individual shipments, plant to demand is not always correct. Often the requirements of multiple markets may be placed on a single truck for distribution via a tour. Models of this situation are now being formulated and solved (Berger et al., 1997). Areas of interest not mentioned in the ReVelle-Laporte article include (1) capacitated plant location, (2) the cost function of the plant, and (3) integrated plant location. In the past dozen years, investigators have focused more on capacitated plant location, often using Lagrangean relaxation. Beginning with the reports of success by Davis and Ray (1969), we have had conflicting messages. Often the messages are anecdotal. Davis and Ray claimed their linear-programming-equivalent methodology produced no fractional variables, whereas others found many fractions. It appears that the relation of the sum of capacities to sum of demands has a significant effect. Heller et al. (1989) and Rosing (1981) both find that as the sum of capacities approach within 3O%_t of the sum of demands, fractional facility variables increase in frequency. This little noticed phenomenon means that the successes reported by some researchers may be data-set dependent. New work, exploring the full range of ratios of capacity to demand, is needed. The concept of Heuristic Concentration (Rosing and ReVelle, 1997) could have a role in resolving some of these problems by reducing the solution space via heuristics, and then applying an exact method. The cost function of the plant is strongly related to the issue of capacity. A capacitated plant may be interpreted as a plant for which the cost of increasing production beyond some level becomes excessive. That is, a convex portion of the cost curve makes further production uneconomical. Thus, if after the fixed charge portion of the cost curve, a convex cost function can be successfully modeled, the issue of capacity could become irrelevant - in cases where capacity really means excessive cost of expansion. Finally, on the plant location side, integrated plant location is little explored. By integrated’ I mean that the components are layered - from the resource nodes to the plant nodes to the warehouse level to the final customer level. The papers which cover all the elements of this area can be counted on one hand. The area of network design or extensive facility location (Mesa and Boffey, 1996) is rich in applications. These include the siting or improvement of highway systems, the location or expansion of rail and transit systems, the design of airline networks, the determination of bus routes, the planning of collection and distribution routes, the construction of power line networks, and of telecommunication networks. Many of these networks have associated point facilities such as stops, hubs, garages, depots, switching stations, computers, concentrators, and the like. There is a richness of applications here. Issues within these areas which are in need of examination include the reliability of networks and the design of tree networks. Reliability of a system may not even be clearly defined mathematically, but reliability of service to a particular node does suggest multiple routes between the source node(s) and the demand node. As a consequence, the network designed to achieved some reliability standard is unlikely to be a tree. And even if a network designer had a problem statement in which a tree would do, it is not clear whether the tree ought to be a subtree of the minimal spanning tree, or any subtree which satisfies the constraints and optimizes the objective. For now, most applications I know of argue for and develop a subtree of the minimal spanning tree. It is easy to argue for the subtree of the MST, among other reasons because the more relaxed problem is unsolved, to my knowledge.
A perspective on location science
11
Currently, I am, among other areas, interested (with my current and former students and collegues) in two related areas: (1) in the spatial and temporal management of forests for timber and wildlife purposes, and (2) in natural area planning in which parcels are assembled or connected into systems for the protection of multiple species. (See, for instance, Church et al., 1996; Snyder and ReVelle, 1996a,b; Snyder and ReVelle, 1997; Williams and ReVelle, 1996, 1997.) The combinatorial aspects of location problems are displayed here and the further challenge of temporal decisions and spatial compactness of assemblages make the problems fascinating. The need for networks of assemblages and road networks in forest management make the problems more interesting yet. I am also interested in emergency services which combine both ground and air transportation for victims of trauma. These problems are complicated by the presence of interacting services and facilities - helicopters and trauma centers. Finally, I think there is opportunity in two related problem settings - the clustering problem and the quadratic assignment problem. The clustering problem seeks the ‘natural clusters’ of data or areas. The goodness of the clustering involves either the nearness of members of the clusters to each other or the nearness of members to some central member of the cluster or the tightness or area of the polygon enclosing the members in each cluster. Obviously, the second of these is a p-median style problem. The first of these is akin to the objective of the quadratic assignment problem (QAP) in that the nearness of two nodes is only counted if the two nodes are in the same cluster - just as in the QAP where the interaction is weighted by the distance between the activities, which is determined by their placement in the grid. The QAP is both a problem from industry and a problem in the spatial planning of metropolitan regions. My hunch is that the formulation or heuristic that solves one (clustering or the QAP) will also, suitably modified, solve the other. How do we foster the growth of our discipline? The simple answer, I think, is by being relevant, by attacking real problems. But we could use, in addition, a means to facilitate our research. I would like to suggest that all members of SOLA or readers of this piece who are US citizens write to Dr. Neal F. Lane, Director of the US National Science Foundation. Urge Dr. Lane to support the funding of a multi-university consortium center for location scholarship - a Center for Siting and Distribution Research in which scholars from industry and academia interact. Your letter should stress not only the importance of the research to the competitiveness of US industry and commerce - with as much detail as possible - but also the importance of our research to governmental institutions and the public sector, including the military, the post office, hospital and trauma systems, ambulance and fire systems, national forest management, regional management of water pollution, solid waste management, school systems, and species preservation through land set aside. Urge that the center should have wide membership and should bring scholars from overseas to interact with American scholars. Suggest also that the center have an advisory board that reflects society’s locational needs including industrialists, heads of store chains, hospital systems administrators, school system administrators, public works directors and the like. The address is Dr. N.F. Lane, Director, US National Science Foundation, 4201 Wilson Boulevard, Arlington, VA 22230. Even if such a center is never created, we will continue to be vital and productive because we offer alternatives to real problems that people face daily. People make siting decisions every day, important decisions that influence the profits of industry, life-and-death decisions for emergency services, bottom-line decisions for retail firms, and strategic decisions for
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CHARLES
REVELLE
defense. With an agenda like this, and with a dedicated our success is assured.
and cooperating
core of researchers,
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A perspective
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