A multivariant travelling-salesman problem

U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

.,Vo1.25,No.Z,pp.200-

201,1985

0041-5553/85 $10.00+0.~ PergamOn Journals Ltd.

A MULTIVARIANT TRAVELLING-SALESMANPROBLEM* A-0. ALEKBEEV

A multivariant travelling-salesman problem is formulated, iteration algorithm to solve it is proposed.

and an

1. Statement of the problem. The present paper discusses a multivariant travelling-salesman problem where several possible versions of the travel between locations are specified, for example journeys by car, rail or plane. Each version is characterized by the travel duration and the cost. It is required to specify the salesman's itinerary with the minimum travelling time, so that the total cost does not exceed a specified sum. The multivariant problem in question belongs to the class of minimax problems, and can be formulated as follows. It is required to determine T= min mar fil(k,), * where

(4.0

n=((ri)lr,,>O), (,,=l. 2, ....L,bi-0,1. ....u.i-0, 1. ._.,,I under the constraints

(1.2)

(1.3)

(1.4)

(1.5) ci, =z

i,if location j is visited

after

location

i,

{ 0 otherwise.

Here &J is the number of the version of travelling from the i-th location to the j-th; i-th location to the ~,,(1,,),c,.(1,,) are the time and cost associated with the travel from the u,,U, are arbitrary real values, and c is the I,,-thvariant; j-th, which corresponds to the feasible total cost. For Li,-1,1,i-0, i,...,n,when condition (1.4) is ignored, the problem formulated reduces to The following problem of scheduling theory, a minimax travelling-salesman problem (see /l/). which refers to the optimization of continous production, can be reached to problem (1.1) The completion time of the j-th job, (1.5). There are R jobs for producing an article. In addition, the completion time of the j-th job (8, 1 depends on the preceding i-th job. depends on the technical version used, the version being described by the cost of carrying it out. In continuous production the output is determined cycle, which is found from the expression

by the duration

of the conveyor's

T=maX[l,,((i/) Ir,,>O)I. Thus, when the total cost is The maximum output is secured with a minimum production cycle. limited, the problem of optimizing the output of a continuous line reduces to problem (1.1) (1.5). In general, the jobs should be performed in a definite order, which can be achieved by introducing the corresponding prohibitations into the matrix IIL,,II. 2. The algorithm for solving the problem. We shall consider the algorithm for solving problem (1.1) -(1.5), based on successive Its concept is application of the algorithm for the usual travelling-salesman problem. close to that of an algorithm for solving a transport problem by the time criterion (see /2/). Let the inequalities 1,,(1,,)>1,,(&,+ I), c,,(Iz,)~c.,(l,,+l), 1,,-1, 2, ....L,,-i,i,j=o, 1.. ..)

n

hold. we determine problem (k-l::

the matrix

I\#

II-llc,,(l,,=i)/j

*zh.vychisl.Mat.mat.Fiz.,25,4,631-633,1985

and solve the following

2oo

travelling-salesman

201

where

the constraints (1.2)-(1.4) are observed. After solving problem (2.1) with the above constraints, we check if condition (1.5) is satisfied. If it is not, problem (l.l)-(1.5) has not solution. If it is satisfied, we find 1%) (3) (I, Ti'=maz f,,((0)I.fI, >0). '.,=(,,(/,,=I).

Using

condition

1;;'=n,a\'r,,(/,)
'!:~'!I and the corresponding

matrix

iic:?'ll

We again solve problem (2-l), (1.2)-(1.4) when h=2 and perform all the operations described. If condition (1.5) is not satisfied at the k-th iteration, the plan obtained at I$-” the preceding iteration 1, I, corresponds to the optimal solution of problem (l.l)-(1.5) and the quantity T=T(‘-“. The algorithms discussed for solving the multivariant cases use standard computer programs written specifically for travelling-salesman problems. The methods of linear interpolation can be used to reduce the number of iterations in searching for the optimal solution of problem (l.l)-(1.5). Here the order of forming the and lJc.,II is somewhat modified, but the general concept of the algorithm rematrices llll,lt mains unchanged. REFERENCES 1. ALJZKSEEV O.G., The minimax problem of scheduling theory, Avtcmatika i telemekhanika, No.1, 108-111, 1978. 2. ALEKSEEV A.O., A method of increasing the efficiency of alqorlthms for solving a transport problem by the time criterion, Izv. AN SSSR, Tekh. kibernetika, No.1, 34-38, 1982.

Translated

U.S.S.R. Comput.Maths.l~ath.Phys.,Vo1.25,No.2,pp.201-202,1985 Printed in Great Britain

by W.C.

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LETTER TO THE EDITOR* In the paper entitled "Estimation of a function from randomized observations" published in the "Journal of Computational Mathematics Mathematical Physics" ~01.23, No.1, 1983, the author, when determining the order of accuracy of the estimate Z(J.1~.I), incorrectly evaluated the proability of matrix (9) converging to a unit matrix which decreases as m increases. If condition (4) is satisfied, when the number of observations N increases the convergence of all eigenvalues L, I=-& z....,m of matrix (9) to unity can be demonstrated only when m=o(s'-~~~), since the expectation is

Therefore, only the following above paper can be formulated.

assertion,

which

is weaker

th-ir.that of Theorem

3 of the

Under the conditions of Theorem 1, the deviation in the norm L,(pj of the Assertion. I(:(. m. I) from l(r) is probably of order \"-: -2: . when t~xfi'-P~~.Thus, projection estimate for functions j(z)which satisfy condition (2), regarding the order of deviation the estimate ,l.ll.j. For observations with random errors, paragraph 4) of Theorem 5 should read as fOllOWs: ~(3,m. Z) from i(=)is of order ~YP(~-~)"*uT~) L,(P) of the projection estimate the deviation in norm ,,~x.\tz-*r ~zprl)or A%<#-?z 1 if t,c~i. and ,rix\:-~ 2;r,=(?+#-2/,)/(4-2$). if par, , and of order I(Z) which belong to the Sobolev classes II.T(.\,), when the piecewiseFor the functions Tln,*=nl are given in non-intersectinq polynomial base functions Vhl(I).h-l. L....!. ,=I. ".....,ri*. the optimal order C/*X,*=X, so that v*,(z)=u if LX.\,," and JI~~.(~)~~,,(II,L(~~T)=~,~~~, subintervals X..", of decrease *Zh.vycbisf.I,fat.mat.Fiz. ,25,4,634-635,1985