- Email: [email protected]
OR Utopia
European Journal of Operational Research 119 (1999) 224±234
www.elsevier.com/locate/orms
OR Utopia Rainer E. Burkard
*
TU Graz, Institut f ur Mathematik B, Steyrergasse 30, A-8010 Graz, Austria
Abstract An utopian form of the OR community is described. We discuss the fractal borders of OR, show some landscapes like the assignment district and report about the laws, the arts, the social structure and the rules in OR Utopia. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Utopia; Laws; Arts; Ethics
*
E-mail: [email protected]
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 2 7 - 7
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
225
1. OR Utopia Sir Thomas More (Fig. 1), lord chancellor of England, humanist and martyr, conceived in the summer of the year 1515 Utopia, a discourse on the ideal commonwealth. In my re¯ections I would like to draw a picture of Utopian forms of our community which I call OR Utopia. The name Utopia stems from a play of words between the Greek ot ± not and et ± good. Let me cite a poem from the preface of the ®rst issues of Thomas More's Utopia:
Fig. 1. Thomas More (1478±1535) as lord chancellor. Painting by Hans Holbein the Younger (1527) ± New York, Frick Collection.
226
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
Fig. 2. Wood-engraving by Ambrosius Holbein from the print of Utopia, Basel 1518.
Citing Oscar Wilde we can say: ``A map of the world that does not include Utopia is not worth even glancing at, for it leaves out the one country at which Humanity is always landing. Progress is the realization of Utopia'' [9]. In 1516 the Dutch painter Ambrosius Holbein, brother of the famous Hans Holbein the Younger depicts Utopia as an island (cf. Fig. 2). And we describe OR Utopia as a land within the wide scienti®c community. Fractal is the structure of OR (Utopia). Can we see borders, can we de®ne OR in a sharp way? Many eorts were made in the past decades to de®ne and describe OR as a closed community. But if we take a closer look the shapes reveal their fractal structures. Great things cannot be de®ned in a strict way, or can anybody de®ne wisdom or love? And the same holds for OR: it has many facets and faces and every de®nition just catches one dimension. The borders of OR Utopia have yet another quality: people can come and go, without passport or bodycheck. There is no quota for foreigners as in many Western European countries. All are welcome. Let us take a close look at a part of the border of OR Utopia where three countries meet: it is a peaceful border between OR, mathematics and computer science (see Fig. 3). There are many corners like that, between OR and management science, economy, logistics and many, many other domains. As we can see from this picture it is often dicult to say: are we in the ®eld of OR or already in mathematics or in computer science? But is this really essential?
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
227
Fig. 3. Peaceful border between OR, computer science and mathematics. By courtesy of H. Behme.
2. Description of some landscapes in OR Utopia I would like to report to you now about some excursions in this wild and fascinating OR country. One of my ®rst journeys led me in the assignment district. There was a valley, called linear sum assignment and there was another valley, called linear bottleneck assignment, but there was no road and way between these two. We tried to climb the mountain between these two valleys and suddenly we found a path to the top of the ridge, called algebraic assignment peak (see Fig. 4). From there we had a wide view: we
Fig. 4. In the assignment district.
228
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
saw the two valleys, and many more. It oered a marvelous view of the whole district, its roads called algorithms, and its people. All who know me, know that I am a passionate mountaineer: having a wide view oers unbelievable new aspects. And the same is true in OR, when getting a general view. Algebraic assignment problem [6] Let H ; ; be a totally ordered semigroup with composition and order relation . Let cij 2 H for i; j 1; 2 . . . ; n. Find a permutation u of the numbers 1; 2; . . . ; n such that c1u 1 c2u 2 . . . cnu n becomes minimum. If H ; ; is chosen as R; ; 6 we get the sum assignment problem: min u
n X ciu i : i1
If H ; ; is chosen as R; max; 6 we get the bottleneck assignment problem: min max ciu i : u 16i6n
Another time we visited a neighboring district, called quadratic assignment. We walked the hooked roads o and on and followed also the path to in®nity (see Fig. 5). And there, the country became ¯atter and ¯atter, with many local minima ± a feature we could also see in other areas. Theorem ([4]). If the coecients of the quadratic assignment problem are uniformly distributed random variables, then the ratio between the maximum and minimum value of the objective function tends almost surely to 1 as the size of the problem approaches in®nity. My friends and I, we visited many other districts and met there wonderful people, hiking like ourselves. Let me mention Jan Karel Lenstra, who has been honored like me this day, or our Polish expedition under the leadership of Jacek Blazewicz and Jan Weglarz, always on the way in the scheduling district, or my friends and companions on joint excursions, Dominique de Werra, Jakob Krarup and Peter Hammer, just to mention a few of them. I met so many friends, whose friendship and encouragement I would never like to miss. But here I would also like to thank those persons who enabled me to visit this country, above all my
Fig. 5. Rainbow to in®nity. By courtesy of H. Behme.
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
229
family: my wife Heidi and my children Michael, Reinhild and Thomas who were often alone without husband and father, he being on a journey, physically or mentally. I also thank my colleagues and students with whom I shared so many adventures and endeavors and who showed me so much in this country. But let me now report about my recent trip to the fascinating district of special cases of NP-hard optimization problems. This district is located like a chain of pearls in the vast realm of NP-hard optimization problems (see Fig. 6). At ®rst we saw a row of 2n points in regular distances and we were asked: what is the shortest way to connect all these points? The answer is simple, if n is even, said Frank Hwang [7], and showed the solution. And for n odd, he continued, I propose that the solution looks alike. But Thomas Maier used a computer and suddenly found a shorter connection. So what? But ®nally we found the way [3]: two lines had to be drawn to ®nd those Steiner points which lead to the really shortest connection (see Fig. 7). Another time Sir Hamilton asked us to ®nd the shortest path through those tiny villages as shown in Fig. 8. It is easy, said Vladimir Deineko [2], let us check, if the distances cij between the villages i and j ful®ll the following conditions for all i; j; l with i 1 < j; l; j 6 l: ci;i1 cjl 6 cil cj;i1 ; ci1;i cjl 6 ci1;l cji : And these can be checked in O n4 time: And really, they do ful®ll these relaxed Monge conditions which immediately lead to a shortest tour (see Fig. 9).
Fig. 6. Eciently solvable special cases in the realm of NP-hard problems. By courtesy of H.-O. Peitgen.
230
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
Fig. 7. Minimum Steiner trees for ladders.
Fig. 8. Instance of the Euclidean TSP. Taken from Ref. [2].
Fig. 9. Optimal solution for the TSP instance in Fig. 8.
But there are many undiscovered areas: no living being knows how to ®nd eciently the shortest tour in circulant networks [10]. No living being knows, if there is a path between the regions P and NP-hard. So many things are still to be discovered, and every new result opens another wide area for investigations. Many OR Utopians are engaged in improving features and optimizing procedures. Once we were asked: can you help us with the automatic control of vehicles in our automated warehouse? We were shown a big hall with huge cranes and informed about the problem. And all of our team thought about this problem
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
231
Fig. 10. Automated warehouse with three stacker cranes.
and we suggested solutions [5]. In a year we were invited to view the ®nished work: the cranes moved like ghosts, it was as if we climbed another peak (see Fig. 10).
3. Laws, arts and ethics in OR Utopia Let me tell you now in short about the laws, the arts, the social structure and the rules in OR Utopia. As we heard in the poem at the beginning, our country OR Utopia has splendid laws. Indeed, and everybody
232
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
can just suggest a law. No parliament is needed, but she or he has to convince the other citizen about the correctness of the law. Let me just show you one example of a beautiful law [1]. Arrow's Impossibility Law: There is no social welfare function which gives a positive association of social and individual values, is independent of irrelevant alternatives, obeys the citizens' sovereignity and has no dictator. In the neighboring country Mathopia, the situation concerning laws is even more interesting: whereas the vast majority obeys the law that every vector space has a basis, a minority, called intuitionists does not acknowledge this law. They only acknowledge what explicitly can be constructed (a feature which reminds me to unbelieve in Saint Thomas), but there is no quarrel between these two groups. Everybody respects the other. When educating their children, OR Utopians tell them an old story which Socrates taught his students more than 2000 years ago and which we are told in Plato's Phaidros. Once Thamos, king of Upper Egypt got a visit from God Thot, the god of wisdom and inventions. (When the sun disappeared Thot ®lled the dark with his light by the moon.) Thot praised the invention of the art of writing with many words, but King Thamos replied: ``You praise the new inventions with many words, but new inventions make man haughty: they feel superior, look down upon those not knowing the new skills and thus they become a burden for society.'' OR Utopians show a strong social responsibility: they do not expel people who suer from a disease or become old and inactive: no, they take care of them and keep them in good remembrance. Arts play a crucial role in the OR community: life is art and art is life. In an old poem, the great Hua [8] describes the solution for ®nding the 1-median in a graph with the following Chinese poem (see also Fig. 11):
Fig. 11. The solution of the 1-median problem [8].
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
233
Many years ago, there was a big discussion among the citizens of OR Utopia. Some said: let us make rules for our society. And they started to set up rules: · Thou should honor OR. · Thou should attend meetings to learn about recent developments and to get new ideas and make friends. · Thou should honor the old, even if they have become inactive and ill. · Should not publish the results of your students under your name. · · · But then a wise old man stood up and said: we will never be able to settle all situations by such rules. Let us just live according to three principles:
Acknowledgements The ®rst and the last picture as well as the picture in Fig. 6 are taken from the book `The Beauty of Fractals', H.-O. Peitgen and P.H. Richter, Springer Publisher, Berlin, 1986. I greatly appreciate the permission to reproduce these images by H.-O. Peitgen, Bremen. The pictures in Figs. 3 and 5 are taken from the book of M. Schr oder, ``Fractal, Chaos, Power laws. Minutes from an in®nite paradise'', Freeman, New York, 1991. I greatly appreciate the permission to reproduce these images by H. Behme, G ottingen. Fig. 4 shows the Tyrolean Alps after a photo made by M. Burkard. I thank also Lin Yixun for the translation of Hua's poem.
234
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
References [1] K. Arrow, Social Choice and Individual Values, Cowles Commission Monograph 12, Wiley, New York, 1951. [2] R.E. Burkard, V.G. Deineko, On the traveling salesman problem with a relaxed Monge matrix, Information Processing Letters 67 (1998) 231±237. [3] R.E. Burkard, T. Dudas, Th. Maier, Cut and patch Steiner trees for ladders, Discrete Mathematics 161 (1996) 53±61. [4] R.E. Burkard, U. Fincke, Probabilistic asymptotic properties of some combinatorial optimization problems, Discrete Applied Mathematics 12 (1985) 21±29. [5] R.E. Burkard, B. Fruwirth, G. Rote, Vehicle routing in an automated warehouse: Analysis and optimization, Annals of Operations Research 57 (1995) 29±44. [6] R.E. Burkard, W. Hahn, U. Zimmermann, An algebraic approach to assignment problems, Mathematical Programming 17 (1977) 318±327. [7] F.K. Hwang, A primer of the Euclidean Steiner problem, Annals of Operations Research 33 (1991) 73±84. [8] L.K. Hua, Applications of mathematical methods to wheat harvesting, Acta Mathematica Sinica 11 (1961) 63±75. [9] O. Wilde, The soul of man under socialism, in The Posthumous Works of Oscar Wilde, The Albatross Continental Library, Hamburg, Paris, Bologna 1935. [10] Q.F. Yang, R.E. Burkard, E. C ß ela, G.J. Woeginger, Hamiltonian cycles in circulant digraphs with two stripes, Discrete Mathematics 176 (1997) 233±254.
www.elsevier.com/locate/orms
OR Utopia Rainer E. Burkard
*
TU Graz, Institut f ur Mathematik B, Steyrergasse 30, A-8010 Graz, Austria
Abstract An utopian form of the OR community is described. We discuss the fractal borders of OR, show some landscapes like the assignment district and report about the laws, the arts, the social structure and the rules in OR Utopia. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Utopia; Laws; Arts; Ethics
*
E-mail: [email protected]
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 2 7 - 7
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
225
1. OR Utopia Sir Thomas More (Fig. 1), lord chancellor of England, humanist and martyr, conceived in the summer of the year 1515 Utopia, a discourse on the ideal commonwealth. In my re¯ections I would like to draw a picture of Utopian forms of our community which I call OR Utopia. The name Utopia stems from a play of words between the Greek ot ± not and et ± good. Let me cite a poem from the preface of the ®rst issues of Thomas More's Utopia:
Fig. 1. Thomas More (1478±1535) as lord chancellor. Painting by Hans Holbein the Younger (1527) ± New York, Frick Collection.
226
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
Fig. 2. Wood-engraving by Ambrosius Holbein from the print of Utopia, Basel 1518.
Citing Oscar Wilde we can say: ``A map of the world that does not include Utopia is not worth even glancing at, for it leaves out the one country at which Humanity is always landing. Progress is the realization of Utopia'' [9]. In 1516 the Dutch painter Ambrosius Holbein, brother of the famous Hans Holbein the Younger depicts Utopia as an island (cf. Fig. 2). And we describe OR Utopia as a land within the wide scienti®c community. Fractal is the structure of OR (Utopia). Can we see borders, can we de®ne OR in a sharp way? Many eorts were made in the past decades to de®ne and describe OR as a closed community. But if we take a closer look the shapes reveal their fractal structures. Great things cannot be de®ned in a strict way, or can anybody de®ne wisdom or love? And the same holds for OR: it has many facets and faces and every de®nition just catches one dimension. The borders of OR Utopia have yet another quality: people can come and go, without passport or bodycheck. There is no quota for foreigners as in many Western European countries. All are welcome. Let us take a close look at a part of the border of OR Utopia where three countries meet: it is a peaceful border between OR, mathematics and computer science (see Fig. 3). There are many corners like that, between OR and management science, economy, logistics and many, many other domains. As we can see from this picture it is often dicult to say: are we in the ®eld of OR or already in mathematics or in computer science? But is this really essential?
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
227
Fig. 3. Peaceful border between OR, computer science and mathematics. By courtesy of H. Behme.
2. Description of some landscapes in OR Utopia I would like to report to you now about some excursions in this wild and fascinating OR country. One of my ®rst journeys led me in the assignment district. There was a valley, called linear sum assignment and there was another valley, called linear bottleneck assignment, but there was no road and way between these two. We tried to climb the mountain between these two valleys and suddenly we found a path to the top of the ridge, called algebraic assignment peak (see Fig. 4). From there we had a wide view: we
Fig. 4. In the assignment district.
228
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
saw the two valleys, and many more. It oered a marvelous view of the whole district, its roads called algorithms, and its people. All who know me, know that I am a passionate mountaineer: having a wide view oers unbelievable new aspects. And the same is true in OR, when getting a general view. Algebraic assignment problem [6] Let H ; ; be a totally ordered semigroup with composition and order relation . Let cij 2 H for i; j 1; 2 . . . ; n. Find a permutation u of the numbers 1; 2; . . . ; n such that c1u 1 c2u 2 . . . cnu n becomes minimum. If H ; ; is chosen as R; ; 6 we get the sum assignment problem: min u
n X ciu i : i1
If H ; ; is chosen as R; max; 6 we get the bottleneck assignment problem: min max ciu i : u 16i6n
Another time we visited a neighboring district, called quadratic assignment. We walked the hooked roads o and on and followed also the path to in®nity (see Fig. 5). And there, the country became ¯atter and ¯atter, with many local minima ± a feature we could also see in other areas. Theorem ([4]). If the coecients of the quadratic assignment problem are uniformly distributed random variables, then the ratio between the maximum and minimum value of the objective function tends almost surely to 1 as the size of the problem approaches in®nity. My friends and I, we visited many other districts and met there wonderful people, hiking like ourselves. Let me mention Jan Karel Lenstra, who has been honored like me this day, or our Polish expedition under the leadership of Jacek Blazewicz and Jan Weglarz, always on the way in the scheduling district, or my friends and companions on joint excursions, Dominique de Werra, Jakob Krarup and Peter Hammer, just to mention a few of them. I met so many friends, whose friendship and encouragement I would never like to miss. But here I would also like to thank those persons who enabled me to visit this country, above all my
Fig. 5. Rainbow to in®nity. By courtesy of H. Behme.
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
229
family: my wife Heidi and my children Michael, Reinhild and Thomas who were often alone without husband and father, he being on a journey, physically or mentally. I also thank my colleagues and students with whom I shared so many adventures and endeavors and who showed me so much in this country. But let me now report about my recent trip to the fascinating district of special cases of NP-hard optimization problems. This district is located like a chain of pearls in the vast realm of NP-hard optimization problems (see Fig. 6). At ®rst we saw a row of 2n points in regular distances and we were asked: what is the shortest way to connect all these points? The answer is simple, if n is even, said Frank Hwang [7], and showed the solution. And for n odd, he continued, I propose that the solution looks alike. But Thomas Maier used a computer and suddenly found a shorter connection. So what? But ®nally we found the way [3]: two lines had to be drawn to ®nd those Steiner points which lead to the really shortest connection (see Fig. 7). Another time Sir Hamilton asked us to ®nd the shortest path through those tiny villages as shown in Fig. 8. It is easy, said Vladimir Deineko [2], let us check, if the distances cij between the villages i and j ful®ll the following conditions for all i; j; l with i 1 < j; l; j 6 l: ci;i1 cjl 6 cil cj;i1 ; ci1;i cjl 6 ci1;l cji : And these can be checked in O n4 time: And really, they do ful®ll these relaxed Monge conditions which immediately lead to a shortest tour (see Fig. 9).
Fig. 6. Eciently solvable special cases in the realm of NP-hard problems. By courtesy of H.-O. Peitgen.
230
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
Fig. 7. Minimum Steiner trees for ladders.
Fig. 8. Instance of the Euclidean TSP. Taken from Ref. [2].
Fig. 9. Optimal solution for the TSP instance in Fig. 8.
But there are many undiscovered areas: no living being knows how to ®nd eciently the shortest tour in circulant networks [10]. No living being knows, if there is a path between the regions P and NP-hard. So many things are still to be discovered, and every new result opens another wide area for investigations. Many OR Utopians are engaged in improving features and optimizing procedures. Once we were asked: can you help us with the automatic control of vehicles in our automated warehouse? We were shown a big hall with huge cranes and informed about the problem. And all of our team thought about this problem
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
231
Fig. 10. Automated warehouse with three stacker cranes.
and we suggested solutions [5]. In a year we were invited to view the ®nished work: the cranes moved like ghosts, it was as if we climbed another peak (see Fig. 10).
3. Laws, arts and ethics in OR Utopia Let me tell you now in short about the laws, the arts, the social structure and the rules in OR Utopia. As we heard in the poem at the beginning, our country OR Utopia has splendid laws. Indeed, and everybody
232
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
can just suggest a law. No parliament is needed, but she or he has to convince the other citizen about the correctness of the law. Let me just show you one example of a beautiful law [1]. Arrow's Impossibility Law: There is no social welfare function which gives a positive association of social and individual values, is independent of irrelevant alternatives, obeys the citizens' sovereignity and has no dictator. In the neighboring country Mathopia, the situation concerning laws is even more interesting: whereas the vast majority obeys the law that every vector space has a basis, a minority, called intuitionists does not acknowledge this law. They only acknowledge what explicitly can be constructed (a feature which reminds me to unbelieve in Saint Thomas), but there is no quarrel between these two groups. Everybody respects the other. When educating their children, OR Utopians tell them an old story which Socrates taught his students more than 2000 years ago and which we are told in Plato's Phaidros. Once Thamos, king of Upper Egypt got a visit from God Thot, the god of wisdom and inventions. (When the sun disappeared Thot ®lled the dark with his light by the moon.) Thot praised the invention of the art of writing with many words, but King Thamos replied: ``You praise the new inventions with many words, but new inventions make man haughty: they feel superior, look down upon those not knowing the new skills and thus they become a burden for society.'' OR Utopians show a strong social responsibility: they do not expel people who suer from a disease or become old and inactive: no, they take care of them and keep them in good remembrance. Arts play a crucial role in the OR community: life is art and art is life. In an old poem, the great Hua [8] describes the solution for ®nding the 1-median in a graph with the following Chinese poem (see also Fig. 11):
Fig. 11. The solution of the 1-median problem [8].
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
233
Many years ago, there was a big discussion among the citizens of OR Utopia. Some said: let us make rules for our society. And they started to set up rules: · Thou should honor OR. · Thou should attend meetings to learn about recent developments and to get new ideas and make friends. · Thou should honor the old, even if they have become inactive and ill. · Should not publish the results of your students under your name. · · · But then a wise old man stood up and said: we will never be able to settle all situations by such rules. Let us just live according to three principles:
Acknowledgements The ®rst and the last picture as well as the picture in Fig. 6 are taken from the book `The Beauty of Fractals', H.-O. Peitgen and P.H. Richter, Springer Publisher, Berlin, 1986. I greatly appreciate the permission to reproduce these images by H.-O. Peitgen, Bremen. The pictures in Figs. 3 and 5 are taken from the book of M. Schr oder, ``Fractal, Chaos, Power laws. Minutes from an in®nite paradise'', Freeman, New York, 1991. I greatly appreciate the permission to reproduce these images by H. Behme, G ottingen. Fig. 4 shows the Tyrolean Alps after a photo made by M. Burkard. I thank also Lin Yixun for the translation of Hua's poem.
234
R.E. Burkard / European Journal of Operational Research 119 (1999) 224±234
References [1] K. Arrow, Social Choice and Individual Values, Cowles Commission Monograph 12, Wiley, New York, 1951. [2] R.E. Burkard, V.G. Deineko, On the traveling salesman problem with a relaxed Monge matrix, Information Processing Letters 67 (1998) 231±237. [3] R.E. Burkard, T. Dudas, Th. Maier, Cut and patch Steiner trees for ladders, Discrete Mathematics 161 (1996) 53±61. [4] R.E. Burkard, U. Fincke, Probabilistic asymptotic properties of some combinatorial optimization problems, Discrete Applied Mathematics 12 (1985) 21±29. [5] R.E. Burkard, B. Fruwirth, G. Rote, Vehicle routing in an automated warehouse: Analysis and optimization, Annals of Operations Research 57 (1995) 29±44. [6] R.E. Burkard, W. Hahn, U. Zimmermann, An algebraic approach to assignment problems, Mathematical Programming 17 (1977) 318±327. [7] F.K. Hwang, A primer of the Euclidean Steiner problem, Annals of Operations Research 33 (1991) 73±84. [8] L.K. Hua, Applications of mathematical methods to wheat harvesting, Acta Mathematica Sinica 11 (1961) 63±75. [9] O. Wilde, The soul of man under socialism, in The Posthumous Works of Oscar Wilde, The Albatross Continental Library, Hamburg, Paris, Bologna 1935. [10] Q.F. Yang, R.E. Burkard, E. C ß ela, G.J. Woeginger, Hamiltonian cycles in circulant digraphs with two stripes, Discrete Mathematics 176 (1997) 233±254.