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Research Report CCS 382

TILE M-TRAVELLING SALESMEN PROBLEM:A DUALITY BASED BRANCH-AND-BOUNI)

ALGORITHM

by

1. All

J. Kennington*

* v CENTER FORCYBERNETIC

STUDIESThe University of Texas

Austin,Texas 787 12

/1E

80 12 17 018

Research Report CCS 382

THE M-TRAVELLING SALESMEN PROBLEM:A DUALITY BASED BRANCH-AND-BOUND

ALGORITHM

by

I. AliJ. Kennington*

August 1980

*Southern Methodist University

This research was supported in part by the Air Force Office of ScientificResearch under Contract Number AFOSR 77-3151, as part of the first author'sdissertation, and by Office of Naval Research Contract N00014-75-C-0569with the Center for Cybernetic Studies, The University of Texas at Austin.Reproduction in whole or in part is permitted for any purpose of the United

States Government.

CENTER FOR CYBERNETIC STUDIES

A. Charnes, DirectorBusiness-Economics Building, 203EThe University of Texas at Austin

Austin, Texas 78712

(512) 471-1821

ABSTRACT

This paper presents a new model and branch-and-bound algorithm

for the m-travelling salesmen problem. The algorithm uses a Lagrang-

ean relaxation, a subgradient algorithm to solve the Lagrangean dual,

a greedy algorithm for obtaining minimal m-trees, penalties to strength-

en the lower bounds on candidate problems, and a new concept known as

staged optimization. Computational experience for both symmetric and

asymmetric problems having up to 100 cities is presented.

Acesnt.;

NTIC i

.1U 1fl-n)17I0ced 11

Distribut ion/AvaiabiityCodes

DttAvakj and/ol.

ACKNOWLEDGEMENT

This research was supported in part by the Air Force Office of

Scientific Research under Contract Number AFOSR 77-3151. This work

was part of the first author's dissertation.

b

1. OVERVIEW

This paper extends the highly successful algorithm of Held and

Karp [I1, 12] for the travelling salesman problem, to the m-travell-

ing salesmen problem. The algorithm involves the use of a Lagrangean

dual within a branch-and-bound structure. A subgradient procedure

is used to solve the dual and it is shown that a greedy algorithm

may be used to evaluate points of the dual function.

We now formally state the m- trveiw ng zaenmen puotemr. We

define a touA beginning at city n as a sequence of distinct cities

{n, il , i2, ... , i I where k > 1. Given n cities (1, ..., n) and m

salesmen, all based at city n, we wish to find a set of m tours such

that each city other than n is a member of exactly one tour. Let c.cii

denote the distance from city i to city j and let c ni + Cii +1 12

c i+ ci n denote the distance for the tour {n, il, ..., The

objective is to select m tours (one for each salesman) such that the

total distance for all tours is a minimum. A related problem has also

been discussed in the literature in which the objective is to select

at most m tours with total minimum distance (see [2]). Clearly, for

* !m = i both models are the classical travelling salesman problem. If

the distances cij = c for all city pairs, then this model is called

4ymmet)Lic; otherwise, it is called aUymme tic.

This model was first formulated in integer programming terms by

Miller, Tucker, and Zemlin [17]. Svetska and Huckfeldt [18] developed

a specialized branch-and-bound algorithm for this problem which uses a

linear programming relaxation. Also, Gavish and Srikanth [7] developed

i -1-

a branch-and-bound algorithm which uses a different relaxation. The

Gavish-Srikanth model does not permit the use of a greedy algorithm

to solve subproblems whereas our model does. Bellmore and Hong [51

proved that the asymmetric version of this model was equivalent to

)an asymmetric travelling salesman problem on n + m - 1 cities. The

question of whether this problem should be attacked directly as in

[7, 181 and the present work or whether it should be converted to

its one salesman equivalent and solved using [3] or [12] is an open

question.

-2-

II. THE MODEL

In this section we present a new model for the m-travelling

salesmen problem. The model is developed using the notion of an m-

tree, which is a generalization of the Held-Karp 1-tree. The m-

tree is defined such tat it has the marroidal property so that the

integer -rogramming relaxation is solvable via a greedy algorithm.

The original generalization proposed by Held and Karp [11] is not

matroidal.

Let the decision variable

1 if some salesman travelsxij = from city i to city j;

0, otherwise.

Assuming that each salesman is based at city n, we call any tour on

cities 1, ..., n - 1 a subtouAL. Then the m-travelling salesmen pro-

blem may be stated mathematically as follows:

n nmin c,.x. (1)

i=l j=l i' ij

s.t. n 1, for j = 1, n-lIXl = (2)k=l for j = n

n for i = 1, ... , n-l(x Xik = (3)

k=l m, for i = n

x.. = 0or 1 (all i, j)J ! (4)

x = 0 (all i)

no subtours. (5)

For the classical travelling salesman problem (i.e. m = 1), a pair

of expository papers by Held and Karp [11, 12] showed that (4) and

(5) could be replaced by a constraint that required the graphical

structure associated with any feasible vector x to be a 1-tree. A

1-tAee on a graph having n nodes (cities) is a spanning tree on n-1

nodes and two distinct edges connecting node n to two other nodes.

Letting Y = {(x1l, ... , Xln' **, Xnl *... Xnn) : xii = O, xij = 0

or 1 and the edges ij having xij = 1 form a 1-tree}, the travelling

salesman problem may be stated as (1), (2), (3), x c Y, and m = 1.

We now generalize the notion of a 1-tree to an m-tree.

An m-tree on a graph havinig n nodes is an acyPic

graph havig n-rm-1 edges on n-I nodes and 2m edges

such that the edge (n,i) never appeax6 mote btan

twice.

Note that every m-tree has n+m-l edges and an m-tree need not be

connected. A set of 2-trees is illustrated in Figure 1 while Fig-

ure 2 illustrates some graphs having n+m-l, (6). edges which are not

2-trees. The graph in Figure 2a is not acyclic on nodes I and 2

and the graph in Figure 2b has three copies of the edge (5,2).Letting X 4 {(xll, ..., Xln, ... X Xnl, .... Xnn): x =0 ,

x.. = 0 or 1, and the edges ij with xi = 1 form an m-tree}, theIi i

m-travelling salesmen problem may be stated as follows:

min c ijxij (6)

-j = , n-k xk m, for j = n

-4-

1, for i - 1,. n-ik = (8)

k for i = n

x EX (9)

A model having only n constraints in place of (7) and (8) exists

for the symmetric m-travelling salesmen problem (see Ali [1]).

Figures 1 and 2

About Here

I-

ii

I i il"-"it.:':-'' '_..:- -. .... '":":- ..... .. ..INrltll .. .. it11.. . . I1I. ..

III. THE ALGORITHM

In this section we present a branch-and-bound algorithm for the

m-travelling salesmen problem. The algorithm uses a Lagrangean relax-

ation, a subgradient algorithm to solve the Lagrangean dual, a greedy

algorithm for evaluation of a point of the dual function, penalties

to strengthen the lower bounds obtained by the greedy algorithm, and

a new concept known as staged optimization. Each of these techniques

are explained in the subsections to follow.

3.1 Lagrangean Relaxation

The Lagrangean dual of (6) - (9) selected for this algorithm is

as follows:

max 0(u,v) (10)Uv

n-l n-lwhere 0(uv) = min { c. .xi + ui( x** - 1)

XEX1ji=l j =i

n n-l n-l

+ un! Xn- m) + X vj(V x.. - ) +j =i i=l i

nn i=l in

jn-l

After rearranging terms O(u,v) = min {Y .xi} - a where a =

xEX ij 1i k=l(uk + v) - m(un + v ) and c.. = c.. + u. + v. The Lagrangeank k n n ij 13 1

dual has been used by both Held and Karp 111, 12] and Bazaraa and

Goode [3] in their highly successful work on the t-avelling sales-

man problem. Our relaxation is a natural extesiJon of their model.

Lagrangean relaxation for general integer prr;ms has been exten-

sively ;tudied by Geoffrion [101.

-6-

Consider the following results which relate the Lagrangean dual

and the primal.

Theorem 1. (Bazaraa and Shetty [4])

Let x* solve (6) - (9) and let (u*,v*) solve (10), (11). Then

O(u*,v*) < I c..X*..ij 1J 1J

Theorem 2. (Bazaraa and Shetty [4])

e(u,v) is concave.

Theorem 3. (Geoffrion [8])

Let (u,v) be any vectors, let x C X solve C(u,v) and let

x ij - 1, for i = 1 ... , n-i

r

x. - 1, for .j = n,.. -

ixij - 1, for j = 1,n-

xij - m, for j - n.

The vector (r,s) is a subgradient of 0(u,v) at the point (u,v).

The above three results are well-known and are easily proved. Given

the optimal vectors associated with both the primal and dual, the non-

negative quantity

c. .xt. - C(u*,v*)1jL 1j (12)'ii c. x*

i ij j

is called the duatity gap.

-7-

For this work, the subgradient algorithm ([13, 15]) is used to

solve the dual (10) and (11). Our implementation of this general

method is presented below.

ALG-1: SUBGRADIENT OPTIMIZATION METHOD FOR DUAL

0. Parameter Selection and l aitia2ization

a. [Select Step Size Parameters] Select G, H, Xi, .o .

b. [Set Initial Solution] u 0 0, v - 0, u* 0 0, v* - 0,

r* 0 3 s'* - O.

c. [Set Lower Bound] L - -

d. [Obtain Over Estimate] Set C such that C max G(u,v).

11, Ve. [Initialize Counters] i -- 1, h - 0.

1. Solvc SubprLobler

a. [Evaluate Dual At (u,v)] Let x* solve min { c..x..'! and

xtX ij 'J ij

set c.. x* - CX.iJ ij

b. [Determine Subgradient]

- 1, for i = 1, n-

Set r. n

L x* - m, for i=n

•x j - 1, for j = i,1, n-1,

i i. - m, for j = ,

c. [Test For Optimality] I r = s = 0, - p with (u,v) an

optimum for the dual and x* an optimum for the primal.

d. [Improved Solution?] If 0 > L, then L -- , u* u, v* v,

r* - r, s*- s, and go to 2.

-8-

e. (Test Step Size Counter] h h+l. If h = H, go to 3.

2. Move To New Point

(uv) - (uv) + [XI( - 0)/ (r,s) 1] (r,s), and go to 1.

3. Change Step Size Ot TermZnate

If i = G, terminate with (u*,v*) as an optimum for the dual;

otherwise, h - 0, i - i+l, (u,v) - (u*,v*) + {k.[O - e(u*,v*)]/

I (r*,s*) I}(r*,s*), and go to 1.

An excellent discussion of convergence results for the subgradient

algorithm are given in Helgason [141.

3.2 Matroidal Structure of M-Trees

To implement ALG-1 efficiently, one needs a fast procedure for

solving

min { c..x..}. (13)xEX ij 13

13

In this section we show that X has a matroidal structure and (13) may be

solved by a greedy algorithm.

We now present results from matroid theory which will be used

in the development of an efficient algorithm for (13). Recall that

a rnat oid, M = [Z, p], is a finite set Z and a set 4 of subsets of

Z such that the following axioms hold:

I.

Ml. .

M2. If X t 4' and VC X, then Y c 4.

M3. If U, V c 4 with LIf = JVI +1, then there exists an x E U - V

such that VkJ"x E

Let 2 denote the power set of Z. Then the elements of C2 are

z zcalled Zndependent subsets of 2 and 2 - i are called dependent

subsets. A maximal independent subset is called a bae.

-9-

i .. ... .. ... ..... ...

We now show that a greedy algorithm can be used to find a mini-

mum weight base. Let 3C denote the set of bases for some matroid.

Let w : Z-+R be a weight function and extend this to w : 2 Z _ R as

follows:

w(C) = ! w(e), CZ-Z.

eEC

The minimal weight base problem may be described as follows:

Find X c such that

w(X) = min w(B). (14)Bc

An optimal base X may be obtained by employing the following algo-

rithm.

ALG-2: GREEDVY ALGORITHM FOR A MINI 'AL WEIGHT BASE

0. XO - , i - 1, Y Z.

1. Find x. such that w(x i) = min{w(x) : x c V, {xhXi E }. If

no such x. exists, stop, Xi 1 is an optimum.

2. X. 4 il9j{xi, V - Y - {x.}, i - i+l and go to 1.

Edmonds [6] proved that ALG-2 produces an optimum for (14).

(hTo apply Edmonds result to problem (13) we must show that Xt i,(the set of m-trees) is the set of bases for some matroid. If this

can be shown, then (13) can be solved by the efficient greedy algo-

Let A = {(i,j) : i < i < n-1, i < j n-], } # i}, A = {(i,n),

4 (n,i) : 1 < i < n-l}, and let A = AUA. Then we will call a setA

X -A independent if theedges of X do not form n ,vc je. We will

call a set XC'-A independent if IXI < 2m. Furthermore, X CA will

-10-

X -A

be called independent if IXA A < n + m - 1 and XCA and XU',A are

independent. Let be the set of all independent subsets of A.

Clearly a maximal independent subset of A is an m-tree. Hence, we

need only show that [A, V!] is a matroid to prove that ALG-2 solves

(13).

Before proving that [A, 4] is a matroid we develop the following

* preliminary result.

Theorem 4,

Let G be a graph and let Q be the set of all spanning trees of

G. Let EI( SI and E2 CS2 where SI, S2 E 0 and E 1 1 = ml, JE21 = i 2 ,

1 > m2 . Then there exists an arc e E E1 - E2 such that E2 U {e}CS 3

Proof. Suppose G has n nodes. Then E has n - mI components and E2

has n - m2 components. Suppose E has no arc e = (i,j) such that i1

and j are in different components of E This implies that E1 has

at least n - m 2 components. Then n - m I > n - m < 2 which

contradicts mi > m2 . Therefore, there exists an e C E - E which1 21 2

connects two components of E2 and E 2 U. This completes

the proof of Theorem 4.

Theorem 4 will be used in the proof of the following result.

Theorem 5.

[A, 4)] is a matroid.

Proof. Axioms Ml and M2 hold trivially. Therefore, we must show'U1 U2 ip ith A

that M3 holds. Let U and U E 1 with = ji n A = tit i1, 2 d = 1 2

• Li rC A I = Sit i = 1, 2 and u 2 + 1. Let e E U U

Case 1: ( > s2. If s > S2P then there exists an arc e C [(UI (

A) - U2] for which U2 U {el E p.

f-11

Case 2: (sI < s2) . If sI < s2, then t > t2 + 1. Now (UlnA) and

(U 2n A) have no cycles. By Theorem 4, there exists an e c

(UTnA) - (U1e h A) such that (U2 (A)U{e} has no cycles.

Therefore U2 I eoCU3rem.This completes the proof of Theorem 5.

Therefore the-minimal m-tree problem,

rain {I .. x..},xX ii 1

is solvable via ALG-2.

3.3 Separation

In [2] it is shown that the number of feasible solutions for an

n city asymmetric m-travelling salesmen problem having m salesmen is

, S.

n-i (n-2)!Sm / (m-l)! '(15)

The proof of this result is constructive and shows precisely how one

may generate an enumeration tree for this problem.

The tree is generated in two phases with phase 1 corresponding

ton-l', i (n-2)!to in (15) Lind phase 2 corresponding to(ml) Assume that

the tree is constructed from top down and let the single top node

correspond to level 0. All nodes at level 2 in the enumeration tree

A I will have 2 arcs fixed and the tree has n-1 levels since fixing n-1 arcs

uniquely determines an m-tree. The first phase corresponds to the

levels 1 through m while the second phase corresponds to levels m+l

through n-l. The two phases of the enumeration tree construction

are now given.

Phase i: At level 0, construct n-m new nodes by fixing arcs (n,l),

(n,2), ... , (n,n-m). For the node wfth fixed arc (n,j),

construct n-m+l-j new nodes by fixing arcs (n,j+l), ... ,

(n,n-n+l). For any node at level Z (<m) having arcs

-12-

(n,j1), ... , (n,j£) fixed where jk+l > k construct

n-m+i-j£ new nodes by fixing (n,j£+l), ... , (n, n-m-i).

Phase 2: At level k > m, for any partial solution having t arcs

fixed, there are n-l-k nodes which have no fixed arc

into them. Choose one of these nodes, say q. Then

n-k+m-2 new nodes are constructed by fixing the appro-

priate arcs whose "to" nodes is q. In developing the

enumeration tree, the node (city) selected to create

the new nodes is the one whose component in the sub-

gradient differs from 0 the most.

A partial enumeration tree for six cities and two salesmen is illus-

trated in Figure 3.

Figure 3 About Here

3.4 Problem Selection

Following the notational conventions of Geoffrion and Marsten

[9], we represent the enumeration tree by a set of candidate prob-

lems which are maintained in the candidate list. Let CP. denote a

candidate problem with fixed arcs (il, jl), ... , (it, i2 ). Define

Y. corresponding to CP. as Yi {(xll, ... , x) x = "" = xi

)= . Then CP. is the problem.

!1 1

1, for j = 1, ... n-[ xij!

k m, for j = n

1, for i - 1, ... n-l

k L m, for i=n

x E Xl-Yi .

-13-

4 -

We make use of two relaxations of CP. in the branch-and-bound algo-

rithm. The relaxation Pi is the Lagrangean

dual,

max O(u,v)

where 0(u,v) = min { ijxij} , while CPi is simply G(u,v) =xcXr Y, ij1

min { a -. _l is used only at the initial node in thexcX/" Yi ii j i

enumeration tree, and CF2 is used at all other nodes where (uv) isl

the optimal solution of CP..1

Let v(CPi) denote the optimal objective value of CP2. Then a

lower bound for all nodes constructed from CP. is v(CP.). LetI 1 +

be any descendent of CP. with S, + 1 fixed arcs and let the arcs1

selected by the greedy algorithm for -P2 and -2C+ 1 be given by (ilJl)

.. (i ,j (i l ( and (il,Jl b (i, ) (i ,

(imj ). Since the greedy algorithm was used to obtain

the solution to CP- and theni Ci+l1

c. > c-- fork= +2, ... , n+m. (16)IkJk- ik-lik-

1

Thus, summing (16) we obtain

SC. > - (17)k=Z+2 'k k k=k+2 k-lik-1

Adding c.. + c - c. to both sides of (17), wekl kJ k im+nJ mi+n +l Z+i

1+1 - 1 i + £+1 m+n m+n

Then the right-hand-side of (18) provides a lowcr bound for CPi+I

The candidate problem selected at each iteration is the one with

smallest lower bound.

-14-

3.5 The Algorithm

Using the ideas of the previous sections, we now suirmarize the

new branch-and-bound algorithm for the asymmetric m-travelling sales-

men problem. The algorithm incorporates a new idea which we call

staged optimizatiOl. Following the conventions of [9], let CL de-

note the candidate list and INC denote the objective value of the

incumbent. Let Z* denote an estimate of the value of the optimal

solution. This estimate is based on the observed duality gap and,

of course, may be either larger or smaller than the optimal value.

The branch-and-bound algorithm is executed with Z* playing the usual

role of the incumbent, INC. This may substantially aid the fathoming

routine with the risk of fathoming an optimum. If a feasible solutionA A

is found such that INC < Z*, then the fathoming strategy using Z* was

justified and the algorithm guarantees optimality. If the complete tree

is fathomed without obtaining a feasible solution, then the fathom-

ing strategy was not justified and the optimum has objective value

greater than Z*. For this case Z* is increased and the procedure

is repeated. A similar idea has been reported by Marsten and Morin

[161.

The algorithm incorporating staged optimization follows:

ALG-3: BRANCH-AAW-BOUND METHOD FOR M-TRAVELLING

SALESMEN PROBLEM

0. hm , zation

Choose .. for staged optimiz,,L cn, set t 1 1, and

set INC -

1. Sce'e Vt

Use ALG-1 to solve the dual. Let (u*,v*) denote the optimal

-15-

dual variables and let x* solve min{1 C.X. .1 Set the lowerXi 13 13

bound L - 0(u*,v*) and let (r*,s*) denote the subgradivnt of

G(u*,v*).

2. Test For DuaLity Gap

If r* = s* = 0, stop there is no duality gap and x* solves the

primal; otherwise, set Z* - L(l + t)

3. CoutAuct First Levet In Enumneation Tree

Construct the first n-m nodes in the enumeration tree and place

them in the candidate list.

4. Solve Nue Cojidcdate Probiem

Let Y c CL and let x solve min {C c..x } Let ey = C. X,13 13 ij 1]j

xcxy ij ij

and (r, S y) denote Lhe subgradient.

5. Fathom TetA

If 0y > Z* go to 7.

6. Feasi.biIU y Teat

If ry = sy = 0, x* - x y , INC 4- Oy, Z* <- y, fathom candidate prob-

lems with lower bounds greater than Oy; otherwise go to 9.

7. Tunijnation Tcst

If the candidate list is not empty, then go to 4.

8. Neu, Stage Required?

If INC # Z*, then set t - t + 1, L - Z*, and go to 3; otherwise,

stop with x* as the optimum.

9. Separatc

Separate Y, update the candidate list, and 8- to 4.

Note that ALG-3 can be easily converted into a rethod *o find approxi-

mate solutions. The interval of uncertainty at any point in the proce-

dure is [INC, L1. An additional test in step 6 is required to make this

conversion.

-16-

i lI I 4- I II I I . .. ... . .

IV. COMPUTATIONAL EXPERIENCE

The branch-and-bound procedure, ALG-3, has been coded in stand-

ard FORTRAN for an in-core implementation. The code was initially

tested on randomly generated asymmetric problems on a CDC 6600. The

random number generator employed for the generation of problems is

the one available on the CDC FTN compiler. The range used for gen-

erating the distances was [100, 3400]. The same range was used by

Bazaraa and Goode in their computational work on the travelling

salesman problem [3]. Since the code is designed for an in-core in-

plementation, one of the major problems encountered in the computa-

tional testing was core storage. The size of the candidate list

grows rapidly for the larger problems, n > 50 and the storage re-

quirement soon exceeds the a\iilarle 200K octal words, even with

most of the data packed.

The computational efficiency of the code is sensitive to the

selection of parameters. Bazaraa and Goode observed that there is

a trade-off in the amount of computational effort expended in the

maximization of the dual and the branch-and-bound procedure. The

more time spent in ALG-7, the better the lower bound obtained. We

employed Xi = 2 - in the code and G and H were selected based on

the size of the problem. The subgradient procedure increases the

lower bound L rapidly in the initial iterations and minimal increases

are obtained for E > 10. However, for larger problems, we found it

beneficial to use a larger value for G, even though the relative"1increase in the lower bound is minimal. Further, the choice of the

duality gap estimates i is crucial. If they are chosen too large,

then the candidate list grows rapidly, while if they are too small,

-17-

7-- ... --- '

the number of candidate problems solved increases.

Our initial computational experience centered on evaluating

the behavior of the code to parameter selection and to the type

of problem being solved. Table 1 summarizes computational results

for asymmetric problems for n = 30, 40, 50, and 60 with 1 < m < n/10.

The parameter settings for the first seven sets used were H - 5, G =

10 and .01, = .02, and 3 03. We found that the duality2 3

gaps for problems with n = 50 and 60 were smaller than .005 and

further, the number of subproblems generated with lower bounds with-

in one percent of the solution for the Lagrangean dual was large.

Problem sets 9 - 12 were solved with the duality gap estimates set

to i = .005, D2 = .01, 3 = .015.

The general conclusions which may be drawn from the results of

problem sets 1 - 12 is that the duality gap seems to decrease as m

4-creases. Further, the problems with larger values of m are easier

to solve than those with smaller values of m. However, as the prob-

lem size increases, the number of candidate problems within small

* percentages of the lower bound obtained from the Lagrangean dual

increases. To circumvent the consequential storage problem, we have

odevised an approximation technique which simply consists of termi-

nating the branch-and-bound procedure as soon as a solution is ob-

I tained. The solution so obtained is provable to be within a small

percentage of the optimal solution. Since the staged optimization

technique makes use of estimates of the duality gap of a problem,

it is possible to obtain a suboptimal solution within a known percen-

tage of the true solution by terminating the branch-and-bound pro-

cedure as soon as an m-tour is obtained. This approximate solution

technique was tested on problems for n 60. The results are summarized

-18-

in problem sets 13 - 18 of Table 1. To obtain a tighter lower bound,

= 12 was used with 3I = .005, a2 .01, and 3 02.

Table 1 About Here

Table 2 reports computational results for the solution of 25

100-city problems for which solutions were obtained. There were

25 other problems which terminated due to storage limitations be-

fore locating an m-tour. For the use of the approximate technique,

a better selection procedure, which makes use of the subgradient, can

be devised so that an m-tour can be located before the number of can-

didate problems becomes unmanageable (see [1]).

Table 2 About Here

To gain insight into the solution of symmetric m-travelling

salesmen problems, we attempted to solve 50 100-city problems by

randomly generating such problems. The range on the costs used

was the same as for asymmetric problems. For such problems, the

code was slightly modified so that the subgradient employed had n

rather than 2n components. However, the enumeration tree was not

modified for these problems, nor were other specializations made.

Computational results for 15 problems which were successfully solved

are given in Table 3. The remaining 35 were terminated due to stor-

age limits. The problems which were solved h;d nigligible duality

gaps, and thus solutions were obtained before the candidate list

grew. The computational results indicate that the duality gap for

large problems is exceedingly small. Thus, even though minimal

increases are obtained in the course of final iterations of the

-19-

subgradient procedure for maximizing the dual, it is beneficial to

employ larger values for G.

Table 3 About Here

Table 4 gives results on Euclidean problems obtained by the

use of intercity distances which were provided by the Civil Aero-

nautics Board for 59 cities. Because the code has not been designed

for the solution of such problems specifically, the computational

experience on these problems is not extensive. Rather, the focus

here was to obtain insight into the nature of solutions to such

problems. Five networks were chosen arbitrarily and three problems

were defined for each network. The first four networks have 30

nodes each. Networks III and IV differ only in the choice of the

base node. The solutions to the problem are illustrated in Figures

4 - S. The most interesting inference that may be drawn from the

solutions is the relationship between the duality gaps for the prob-

lems and the solutions. The solutions to the six prcblems on net-

works I and II are intuitively obvious, whereas the solutions to

the problems on networks III and IV are not. Thus, the harder the

problem, the larger the duality gap. Furthermore, note the manner

Nin which solutions to problems on the same network are related. From

the first three networks, it would seem that realistic m-travelling

salesmen problems would have to be further con. Lvraied LC ensure

that each salesman visit at least some minim: umber of cities.

Failing this constraint, solutions of the kind illustrated may be

expected. Whereas, as the number of salesnen i,.creases, as in Fig-

ure 7, the solutions become more meaningful. That s, not all but

-20-

one salesmen travel to only one other cityv. The soltio ns obtalned

for the network on 59 nodes are illustrated in Figure 8. For the

problems on networks III, IV and V, even though the initial estimates

of the upper bounds used were quite close to the true duality gap,

the number of subproblems examined before verification of optimality

is large.

Table 4 About Here

Figures 4 - 8 About Here

Based on our computational experience using the technology de-

veloped in this exposition, we have arrived at the following conclu-

sions:

(i) The duality gap for problems with up to 100

cities is quite small (less than 1%).

(ii) Holding the number of cities constant, the

problems become easier to solve as the num-

ber of salesmen increases and the duality

gap decreases.

(iii) Exact solutions for problems with up to 50

I cities can be obtained routinely with the

ii implementation of the algorithm reported.

(iv) A in-core/out-of-core algorithm using this

technique could be used to solve problems

% having 100 cities.

II -21-

REFERENCES

1. Ali, A. I., "Two Node-Routing Problems," unpublished disserta-

tion, Operations Research Department, Southern Methodist Uni-

versity, Dallas, Texas (1980).

2. Ali, A., and J. Kennington, "The M-Travelling Salesmen Problem,"

Technical Report OR 80016, Department of Operations Research,

Southern Methodist University, Dallas, Texas (July 1980).

3. Bazaraa, M. S., and J. J. Goode, "The Travelling Salesman Prob-

lem: A Duality Approach," Mathematical Programming, 13, (1977),

221-237.

4. Bazaraa, M. S., and C. M. Shetty, Nonlinear Programming: Theory

and Algorithms, John Wiley and Sons, New York, NY, (1979).

5. Bellmore, M., and S. Hong, "Transformation of Multi-Salesmen Prob-

lem to the Standard Travelling Salesmen Problem," Journal of the

Association of Computing Machinery, 21, 3, (1974) 500-504.

6. Edmonds, J., "Matroids and the Greedy Algorithm," Mathematical

Programming, 1, (1971), 127-136.

7. Gavish, B., and K. Srikanth, "Optimal and Sub-Optimal Solution

Methods for the Multiple Travelling Salesman Problem," Presented

at the Joint National TIMS/ORSA Meeting, Washington, D. C. (1980).

8. Geoffrion, A. M., "Duality in Nonlinear Programming: A Simplified

Applications-Oriented Development," SIAM Review, 13, 1, (1971) 1-37.

9. Geoffrion, A. M., and R. E. Marsten, "Integer Programming Algorithms:

A Framework and State-Of-The-Art Survey," N;an :;en .ent Science, 18,

9, (1972), 465-491.

10. Geoffrion, A. M., "Lagrangean Relaxation for Integer Programming."

Mal...hematical Programming Study 2, (1974), 92-114.

-22-

11. Held, M., and R. M. Karp, "The Travelling Salesman Problem and

Minimum Spanning Trees," Operations Research, 18, (1970), 1138-

1162.

12. Held, M., and R. M. Karp, "The Travelling Salesman Problem and

Minimum Spanning Trees: Part II," Mathematical Programming, 1,

(1971), 6-25.

13. Held, M., P. Wolfe, and H. Crowder, "Validation of Subgradient

Optimization," Mathematical Programming, 6, (1974), 225-248.

14. Helgason, R. V., "A Lagrangean Relaxation Approach to the Gen-

eralized Fixed Charge Multicommodity Network Flow Problem,"

unpublished dissertation, Department of Operations Research,

Southern Methodist University (1979).

15. Kennington, J., and R. Helgason, Algorithms for Network Pro-

gramming, John Wiley and Sons, New York, NY, (1980).

16. Marsten, R. and T. Morin, "Branch-And-Bound Methods With Stronger

Than Optimal Bounds," Presented at Joint National Meeting of

TIMS/ORSA in Washington, D. C. (May 1980).

17. Miller, C. E., A. W. Tucker, and R. A. Zemlin, "Integer Programm-

ing Formulation of Traveling Salesman Problems," Journal of the

Association of Computing Machinery, 7 (1960) 326-329.

18. Svetska, J. A., and V. E. Huckfeldt, "Computational Experience

With An M-Salesman Traveling Salesman Algorithm," Management

Science, 19, (1973), 790-799.

-23-

.................................-- '

(a)

(b)

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-24-

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-2 5-

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I -26-

(a)

(b)

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On Network I.

-27-

(a) iIIri/ *

(b)

Figure 5. Illustration of Solutions tc Problems

On Network II.

-28-

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Unclassified

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REPORT DOCUMENTATION PAGE BFRE CNOTRLTINO M

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4 t e. I .,h.f,_4 IS TYPE OF REPORT & PERIOD COvERED

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Ili. DISTRIBUTION STATEMENT (of Ih. abstract entred In Block 20, It different hrem Report)

IS. SUPPLEMENTARY NOTES

It. KEY WORDS (Continue an reverse side If nec.eary and Identify by block numbr)

* M-travelling salesmen problem, branch and bound, Langrangean relaxation,rn-tree, staged optimization, matroid

20. ASSTRACT (Conttnue an reverse aide It neceeseinmd Identify by block number)

This paper presents a new model and branch-and-bound algorithm for ther-travelling salesmen problem. The algorithm uses a Langrangean relaxation,a subgradient algorithm to solve the Langrangean dual, a greedy algorithmfor obtaining minimal r-trees, penalties to strengthen the lower bounds oncandidate problems, and a new concept known as staged optimization. Compu-tational experience for both symmetric and asymmetric problems having upto 100 cities is presented.

D FOAN4 1473 EDITIONOFSIL"v 69 I Unclassified

-~ . -,....J iCURnI", CLASSIFICATION OF TMIS PAGB (000ivi os.

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I 0,1