JNCLASSIFIED. [ined^ervices I echnlcäfln. Reproduced by. DOCUMENT SERVICE CENTER KNOTTBUILDINC DftYTON, 2. OHIO

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1 "HiWUl na < v» JNCLASSIFIED [ined^ervices I echnlcäfln Reproduced by DOCUMENT SERVICE CENTER KNOTTBUILDINC DftYTON, 2. OHIO This document I. th. property of th. United State. Gover^ent **?S^tt raiion of the contract and.hall be returned when «^ r f^lntormation Agency, to the following addre..: Armed '»» ^ÄlV»«. 1 Ohio! Dceument SerTice Center. Knott Building, Dayton 2, Ohio. NCrnCE: WHEN GOVERNl«*! ^ CJ^^ T^USED FOR ANY P ^ OT^ TH AN COWEOTON^WUH THEREBy ^^ GOVERNMENT PROCUREIOWT OPERATICW. ^"^Sgj.'JSl) THE FACT THAT THE MO IIESPONSIBILITY. NOR ANY SSH2S!?ft2mm OSWSY WAY SUPPLIED THE GOA'ERNMENT MAY HAVE TORMULATEE^ FOTMSHE^ g^t TOBE^GARDED BY SAL* DRAWINGS, SPECIFICATIONS OR OTK^^AB NOT ^U«* HOLDER OR ANY OfHER IMPLICATION OR OTHERWISE AS WANYM^R U^NSmG TH*^ TO MANUFACTÜRE) PERSON OR CORPORATION, ORCO^^%^^^^/^AY BE RELATED THERETO. USE OR SELL ANY PATENTED INVENTION THAT MAY in A^X UNCLASSIFIED w

2 ^: ^Ql nz CD oo u. wß4in[/ c T&naject RESEARCH MEMORANDUM This is a working paper. It may be expanded, modified, or withdrawn at any time. The views, conclusions, and recommendations expressed herein do not necessarily reflect the official views or policies of the U. S. Air Force. -7^ K-fl 11 \) ßonjmtUto* CALIFORNIA- SANTA MONICA

3 #* ' US AIR FORCE PROJECT RAND RESEARCH MEMORANDUM /_ SOME EXPERIMENTS ON THE TRAVELINO- SALESMAN PROBLEM by J. T. Robacker L 28 RM-I52I July 19 b5 J Assigned to This is a working pape:. It may be expanded, modified, or withdrawn at any time. The views, conclusions, and recommendations expressed herein do not necessarily reflect the official views or policies of the United States Air Force. mv 1700 MAIN it. SANTA MONICA CAUfOINIA' 2 5

4 % RII-I52I -1- SUMHARY This paper presents the results of a series of experiments on the traveling-salesman problem. The purpose of these experiments was to Investigate the efficiency of the llnear^rogrammlng technique In solving this type of problem.

5 RK SOMB KXPBRDgliTS ON THB TRAVELING-SAmaMAW PROBLEM J. T. Robacker 1. INTRODUCTION AND DISCUSSION OF RESULTS During th«early part of 195^, Dantzlg, Pulkerson, and Johnson formulated a linear-programming teohnlque for the solution of the traveling-salesman problem [l]. The methods employed by these authors qulokly generated optimal tours when a good tour was known. The question naturally arose concerning the efficiency of the technique when a random tour was chosen as the starting point. Dr. I. Heller conjectured that It might be efficient and proposed that experiments be made as one means tc Investigate this question and to gain a greater degree of Insight into the dynamics of the technique Itself. A series of 10 nlne-clty traveling-salesman problems was considered. In each case the distances between cities were chosen from a table of two-digit random numbers. The starting tour was then chosen to be In the order of the natural numbers (with the cities numbered 1 through 9 at the outset). In the last problem, the technique was applied first to find the maximal tour and then to proceed from it to the minimal tour.

6 -M» "^1 RH Tables 1 to 10 are the distance tables for the 10 problems. An element a 1J (a 1J - a^) In the distance table represents the distance from the 1-th city to the J-th city. It Is to be noted that since these distances were chosen from a table of random numbers, they cannot be construed necessarily to represent straight-line distances In Euclidean space. At the foot of each distance table Is a table listing the tours obtained at the end of each iteration* (the last entry being the optimal tour). Table 11 lists the tours obtained by starting with the maximal tour associated with Table 10 and proceeding to the minimal tour. The largest number of Iterations needed was six, while the average was only a little under four. The average ime to work one of the problems was about 3 hours. Since for a nine-city problem there are ^ - 20,160 possible tours. It is apparent that the simplex method which was employed was extremely efficient. In addition, it is noteworthy that the only secondary constraints introduced in the solution of these problems were upper bounds. In connection with these experiments, A. W. Boldyreff suggested am approximation procedure, the merit of which One iteration is defined to be the process of going from one tour to another tour of the same or shorter length.

7 ^ RM «8 In its Inherent simplicity and in the rapidity with which it may be applied. An application of thie* approximation method to the 49-city problem of \)L] gave a tour of 851 units as compared with the optimal of 699 unit», an error of 21%. 2. BRIEF REVIBW OP THE TICHNIQOB OP SOLUTION The mathematical technique found In [l] will briefly be reviewed here, and will then be Illustrated by way of an example. The factors T., referred to as "potentials," are computed from the formula (1) T. + T - a.. «0 for all (i,j) in the basis. i.e., for all (l,j) corresponding to column vectors in the basis. Computationally, this means first finding a loop with an odd number of links which correspond to vectors in the basis; then starting at a city k of the loop, adding and subtracting the lengths of the links alternately; the resulting number is then 2r.. T* from (1). Having T W we can then compute all the k 1

8 '»J» RM-1521 TT, = -30 IT Z = 7rc 3>V TTg' "^4 = 34 7 =7r 59= TTg" 5jir 5 =20 Figure 1 We next choose the link (k,i) such that (2) T k + T i - Ä kl max (T^ + ^J ~ a ij^ > 0 and allow the value of the corresponding variable x^ to be equal to 0. The values of the basic variables must now be corrected so that the sum of the values of link variables around any city must equal 2. For this purpose we try to find an even loop containing (k,i). For example, suppose the starting tour **

9 r RM~1521 ^ 1» Illustrated as l.i Figure 1 along with the potentials and links corresponding to basic variables. The numbers on the links are the lengths of those links as chosen from Table 1. It Is seen that the greatest value which (6,2) may have Is 0 since the value of each link In the basis Is bounded above by 1. Consequently, we put a bar on (9.1), Indicating that It has attained Its upper bound and Is replaced In the basis by (6,2). Since (9,1) Is no longer In the basis, we must recompute the prices according to (l). Figures 1 5 Illustrate the steps necessary to complete the first Iteration. The solid lines are those In the tour. * In Figure 5 a new tour Is achieved. By repeating the process we ultimately reach the optimal tour when (3) "'l "*" T J ~ a lj ^ 0 for a11 1 ',J " 1 ' 2 '--" 9 - t

10 . ^ ^ RK-1521 Figure 2 I

11 RM Plgure 3 Figure 4

12 n RM »- Figure 5 I

13 ,, ^ '^0^ RH-1521 _8_ 3. AN APPROXIMATION TECHNIQUE The following approximation procedure has been suggested by A. W. Boldyreff. Start with the first three cities. There Is then only one tour of these, namely, 123. We now set (4) F(1,J;M - a l4 a j4 - a 1J 1,J - 1.2,5 Let k and i be the cities such that (5) P(k,i;4) - mln P(l,Ji4) 1,J-1,2,5 Then our tour of the first four cities replaces the link (k,l) in the tour of the first three cities by the two links (k,4) and (4,1). We repeat this process, adding one city at a time, until we have a tour of all the cities. To illustrate let us consider Table 1 and tabulate the procedure as follows.

14 I X city % l«n«th of link & >< 3 * 1 city 4 J^ length of link ä 15 i 2 5^ 555 ^ i clt y length of link * l6! X 1 city longth of link * x7 2 5 X city ! «« of llnk a l8 n X 1 city ***** » l9 of link T 9! f S-"* 1 tour longth of link :. At each Btago th. X «arr«th. link which i. replaced, in thi. example the final tour ha. a length of 281 while the optimal tour has a length of 232. The error i. 21^. Inmost

15 i RM case 8 considered here the error 1. lee» than thle, as can be seen from the comparison In the tables of Section 5- There are many ways In which the accuracy of this technique may be Improved, but It Is felt that the resulting complexity would probably «suit In a relatively -11 increase In accuracy.

16 1 IW ü 8 Iteration Tour Length of Tour 0 123^ ^ Approximation Technique * Table 1

17 «SUÄP" 1 RM N u 1 \ 2 96 \ 3 19 ^9 \ \ \ \ \ \ \ City Iteration Approximation Technique Tour ^ * * Table Length of Tour ,.. w>.

18 RJ \ 2 12 \ \ \ \ \ \ \ City Iteration Approximation Technique Tour? Table 3 Length of "Sour _ L.121 m 216

19 R»-1521 Iteration Tour Length of Tour pp ro xlma 11on I echnlque Table

20 n RM c ( Iteration Approximation Technique Tour l2q8t 6^^3 T24567B ^ Table 5 l_l Length of Tour Q

21 It RM ~ 1 \ 2 47 \ \ k \ O \ \ \ K \ V _L ö City 7 55 \ 8 9 Iteration Tour Length of Tour : ' i; Upproximatlon. g vn^^v^v^-l /nie» rxovtuw Ta.ble 6

22 i uz. i± n -17=^ 4 H o I 72 I City Length of Tour 0 I pproximation echnlqu«tftble 7

23 I Rpproxi pproxln» JTechnlc chnique Table S

24 r- IW Ö o Tour 2 3* * 5 & ^ " ^ ipproxinätlonj *5 technique Length of Tour Table 9 (

25 IT"' c ^ RPi o Table 10

26 RM Dlatance Titbl«a*»B in Table 10 o Iteration 2 and 3 Tour * Length of Tour Table 11

27 ilr ~~ r RM RSPERKNCES M Dantzl«. 0., D. Fulkerson, and S. Johnaon, Solution of a Largo Seal«Traveling Salesman Problem, Jour, of the Oper. Ree. 80c. of Amer., Vol. 2, No. 4, ffov., 1^4, pp. jy.muu.

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