17
~' ~"~ JOURNAL :,..,,.. .-:. EUROPEAN OF OPERATIONAL RESEARCH ELSEVIER European Journalof OperationalResearch 105 (I 998) 552-568 Theory and Methodology Designing reliable tree networks with two cable technologies Luis Gouveia ~'*, Eric Janssen b.I a DEIO.CIO, Faculdade de Ci~ncias da Uni~'ersidade de Lisboa, Bloco C/2-Campo Grande, Cidade Unicersitaria, 1700 Lisboa, Portugal b MABES-FEW, Erasmus UniL'ersityRotterdam, Burgemeester Oudlaan 50, 3062 PA Rotterdam, The Netherlands Received 4 May 1996; accepted 21 January 1997 Abstract In this paper we introduce a minimal spanning tree problem with generalized hop constraints and primary connectivity constraints. The problem is concerned with reliability requirements in a centralized network design problem where two different cable technologies are available. The problem is shown to be NP-hard and as a consequence we derive lower bounding and upper bounding schemes for the problem. We formulate the problem as a directed multicommodity flow model. Due to the size of the corresponding LP model we use Lagrangean relaxation together with subgradient optimization to derive lower bounds. A Lagrangean heuristic is developed to construct feasible solutions. The theoretically best lower bound associated to the La~angean scheme quite often improves on the value of the corresponding LP bound since the relaxed problem does not satisfy the integrality property. In fact, using the heuristic, these bounds can be proved to be even optimal for many of the cases tested. These results are taken from a set of complete graphs with up to 40 nodes. We also examine a few variations of the main model. In particular we shall discuss several different ways of modeling the primary connectivity constraints. One outcome of our discussion is that we shall derive an extended and compact representation of the convex hull of directed rooted subtrees when the underlying graph is series-parallel. 1998 Elsevier Science B.V. Ke)words: Multicommodity flows; Trees; ttop constraints;Reliability 1. Introduction This paper addresses a variation of the terminal layout problem (TLP), which consists of linking n terminals at different locations to a central computer site. The n terminals or nodes could also be telephone switches and the central computer site or the root node could also be a telephone centre. By linking all terminals to the computer site in a tree-like topology, the site can send messages to (or receive from) any terminal via the unique path from the site to this terminal. Linking all terminals to the computer centre with minimum cost leads to the well-known and efficiently solvable minimal spanning tree problem (see e.g. Prim, 1957). However, in the context of telecommunications networks, the terminal network involves certain constraints which are either related with the performance of the corresponding network or with the availability of some classes of devices. Corresponding author. E-mail: [email protected]. Currently at Exact Software, Poortweg 6, 2612 PA Delft, The Netherlands. 0377-2217/98/S19.00 1998 Elsevier Science B.V. All rights reser,'ed. PH S0377-2217(97)00067-2

Designing reliable tree networks with two cable technologies

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~' ~"~ JOURNAL :,..,,.. .-:. EUROPEAN

OF OPERATIONAL RESEARCH

ELSEVIER European Journal of Operational Research 105 (I 998) 552-568

Theory and Methodology

Designing reliable tree networks with two cable technologies

Luis Gouveia ~'*, Eric Janssen b.I

a DEIO.CIO, Faculdade de Ci~ncias da Uni~'ersidade de Lisboa, Bloco C/2-Campo Grande, Cidade Unicersitaria, 1700 Lisboa, Portugal b MABES-FEW, Erasmus UniL'ersity Rotterdam, Burgemeester Oudlaan 50, 3062 PA Rotterdam, The Netherlands

Received 4 May 1996; accepted 21 January 1997

Abstract

In this paper we introduce a minimal spanning tree problem with generalized hop constraints and primary connectivity constraints. The problem is concerned with reliability requirements in a centralized network design problem where two different cable technologies are available. The problem is shown to be NP-hard and as a consequence we derive lower bounding and upper bounding schemes for the problem. We formulate the problem as a directed multicommodity flow model. Due to the size of the corresponding LP model we use Lagrangean relaxation together with subgradient optimization to derive lower bounds. A Lagrangean heuristic is developed to construct feasible solutions. The theoretically best lower bound associated to the La~angean scheme quite often improves on the value of the corresponding LP bound since the relaxed problem does not satisfy the integrality property. In fact, using the heuristic, these bounds can be proved to be even optimal for many of the cases tested. These results are taken from a set of complete graphs with up to 40 nodes. We also examine a few variations of the main model. In particular we shall discuss several different ways of modeling the primary connectivity constraints. One outcome of our discussion is that we shall derive an extended and compact representation of the convex hull of directed rooted subtrees when the underlying graph is series-parallel.�9 1998 Elsevier Science B.V.

Ke)words: Multicommodity flows; Trees; ttop constraints; Reliability

1. In t roduc t ion

This paper addresses a variation of the terminal layout problem (TLP), which consists of linking n terminals at different locations to a central computer site. The n terminals or nodes could also be telephone switches and the central computer site or the root node could also be a telephone centre. By linking all terminals to the computer site in a tree-like topology, the site can send messages to (or receive from) any terminal via the unique path from the site to this terminal. Linking all terminals to the computer centre with minimum cost leads to the well-known and efficiently solvable minimal spanning tree problem (see e.g. Prim, 1957). However, in the context of telecommunications networks, the terminal network involves certain constraints which are either related with the performance of the corresponding network or with the availability of some classes of devices.

�9 Corresponding author. E-mail: [email protected]. Currently at Exact Software, Poortweg 6, 2612 PA Delft, The Netherlands.

0377-2217/98/S19.00 �9 1998 Elsevier Science B.V. All rights reser,'ed. PH S0377-2217(97)00067-2

L. Goureia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568 553

One type of such constraints are the hop constraints (see, e.g. Gouveia, 1995, 1996) which limit the number of links (hops) between the root node and any terminal and which measure in a certain way the reliability of the tree network. As far as we know, LeBlanc and Reddoch (1990) were the first to suggest hop constraints as a surrogate for reliability in telecommunication networks. Additional applications of hop constraints can be seen in the more recent LeBlanc et al. (1995). Hop constraints have also been used by Balakrishnan and Altinkemer (1992), to generate networks with different costs and performance levels.

In this paper we study a generalization of the TLP where two types of cables with different reliabilities are available. For each connection we can either install a cable of type I, with reliability (1 - a t ) (i.e., with probability (1 - a l) of being operative or equivalently with failure probability a~) or install a cable of type 2 with reliability (1 - a 2 ) . Clearly, the more reliable link is more expensive. Requiring that paths which can contain links with two different reliabilities (1 - a 1) and (1 - a2), have at least a probability /3 of being in operation, corresponds to guaranteeing that (1 - a l ) kt �9 (1 - a2) k2 >/3, where kl and k2 are the number of links of each type in the path. Taking logarithms and changing signs shows that such an inequality is equivalent to kl �9 ln(1 - a l ) -1 + k2 . ln(1 - a2) -1 < In(/3) - I , where In (a ) denotes the natural logarithm of a (0 < a < 1). This indicates that reliability requirements with two types of links correspond to limiting a weighted number of links in each path. These two types of links may refer to two different cable technologies, one more reliable than the other, for example, fibre optic and copper. They could also represent the same kind of cable technology, where the more reliable links refer to cables installed underground and the other to aerial cables. From now on we shall follow the notation given in Balakrishnan et al. (1994) and will denote the two kinds of links by primary (or technology/type I) links and secondary (or technology/type 2) links. Primary links are assumed to be more expensive and to have higher reliability than secondary links.

We also require that primary links have to be connected to the root by a path consisting of primary links only. Such constraints will be referred to as 'primary connectivity' constraints. The reason for using such constraints is that when a message flows from one technology link to another technology link it has to undergo some kind of data transformation, which implies that a switching device has to appear at every node where a change of technology takes place. De Jongh et al. (1995) discuss a different problem with two cable technologies and involving so-called 'transition costs' related to the use of such data transformation devices. In our problem, requiring primary links to be connected to the root node by a path consisting only of primary links keeps the number of such devices small. In fact, with our assumption, at most one switching device is needed in any path from the root node to any other node. Another reason for requiring the primary connectivity constraints is that they imply that more paths can benefit from the higher quality of a primary link.

In graph theoretical terms, our problem, which will be denoted by the Hop and Primary Connectivity Constrained Two Level Minimal Spanning Tree (HPCTLMST) problem is the following:

Consider a graph G = ( V , E ) where V = {0,1 . . . . . n}, with primary and secondary costs c~i and c~j respectively for each edge {i,j} ~ E and natural numbers H, w, and w 2. We want to find the minimal spanning tree where each edge is either primary or secondary and such that: (i) for each node k ~ V\{0} the unique path from node 0 (the root) to node k contains a weighted number of primary and secondary edges (with weights w~ and w 2 respectively) which does not exceed H; (ii) the primary edges constitute a connected subset, containing node 0.

It will be assumed for all {i,j} ~ E that ci. / > c~j and w~ < w 2. Notice that requiring H, w~ and w 2 to be integer is no loss of generality since when these parameters are rational they can be multiplied by the same number to become integer. Also, without loss of generality it can be assumed that H is an integer combination of w I and w 2. The HPCTLMST problem is a generalization of the Hop Constrained Minimal Spanning Tree (HMST) problem, which considers only one type of links (see Gouveia, 1995, 1996). Note that if w I = w z or c~j = cZj for all {i,j} ~ E the HPCTLMST problem reduces to the HMST problem. Gouveia (1995) showed that the HMST problem is NP-hard. Since our problem is a generalization of the HMST problem, it is also NP-hard, which implies that it is very unlikely to find exact and efficient methods for the HPCTLMST problem. Therefore, we develop a heuristic to produce feasible solutions for our problem. To assess the quality of such

554 L. Gout.eia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568

feasible solutions we also d.evelop a lower bounding method derived from an integer programming formulation for the problem.

The remainder of this paper is organized as follows. In Section 2 we formulate the HPCTLMST problem as a directed multicommodity flow model. Several different ways of formulating the primary connectivity require- ments are discussed and compared in Section 3. One set of valid inequalities for the directed model is presented in Section 4 which improve the bounds derived from the linear programming (LP) relaxation. One way of deriving a lower bound for the optimal value of the HPCTLMST problem is by solving the corresponding LP relaxation with or without the extra inequalities. However, solving such LP models takes huge amounts of CPU-time because many constraints are involved. Therefore we resort to a Lagrangean relaxation based scheme which is presented in Section 5. In Section 6 we present a heuristic which provides feasible solutions for the original problem and is embedded in the Lagrangean based scheme. The quality of such feasible solutions can be evaluated by means of the generated lower bounds. In Section 7 we present a simple test that enables us to reduce the size of the problem. Computational results given by both the LP and the Lagrangean relaxations are presented in Section 8 for various randomly generated problem instances. At the end we present a few conclusions.

2. A mult icommodity flow formulation for the H P C T L M S T problem

It is well-known that for several network design problems, better formulations (i.e. more compact formula- tions and /o r formulations with a better LP bound) can be obtained by formulating the problem in a directed graph. See for instance, Goemans and Myung (1993), Balakrishnan et al. (1994) and Magnanti and Wolsey (1995). See also Gouveia (1996) for a similar discussion concerning the HMST problem. Therefore, while the HPCTLMST problem was previously defined in an undirected graph, the models described in this paper are defined in a directed graph. An undirected HPCTLMST instance can be simply transformed into an equivalent directed HPCTLMST model if we replace each edge {i,j} of the undirected graph with two directed arc ( i , j ) and ( j , i ) and the cost of these two arcs is equal to the cost of the original edge. We also assume that the arcs are directed outward from the root and any edge {0,i} (i = 1 . . . . . n) is replaced by one single arc (0,i).

In a network design problem defined in a directed graph G = (V,A) which only involves one single cable technology, every element of the set of feasible solutions is determined by selecting one of two options for every arc in A: whether or not the arc is included in the directed tree. On the other hand, in a network design problem which involves two cable technologies, we have to make a decision between three options for any given arc: whether or not the arc is selected in the directed tree and, if so, what type of cable is selected. One binary design variable is no longer sufficient to represent the information attached to every arc. Following Balakrishnan et al. (1994), we introduce a pair of binary design variables (X i~ ,X 2) for every arc ( i , j ) in A. The first variable indicates whether primary arc ( i , j ) is selected and the second indicates whether secondary arc ( i , j )

( y i p y2p ) for all ( i , j ) ~ A and p = 1 . . . . . n. The first is selected. We also introduce a pair of flow variables - - i j ,-ij - variable indicates whether primary arc ( i , j ) is in the unique path between node 0 and node p, and the second indicates whether secondary arc ( i , j ) is in the unique path between node 0 and node p. Contrary to the model given in Balakrishnan et al. (1994), our model uses two sets of flow variables. The reason for this is that we could not model the generalized hop constraints with only one single set of flow variables. A list containing all variables involved in our directed muliticommodity flow formulation, denoted by DMF, is given next.

Xil~ = 1 ( ( i , j ) ~ A) if and only if primary arc ( i , j ) appears in the optimal solution and Xi~ = 0 otherwise; XI~ = 1 (( i , j ) ~ A) if and only if secondary arc ( i , j ) appears in the optimal solution and X 2 = 0 otherwise; Yi (p = 1 ( ( i , j ) E A; p = 1 . . . . . n) if and only if primary arc ( i , j ) is used in the path from the root to node p and Yi! p = 0 otherwise; Yi~ p = 1 ((i , j ) ~ A; p = 1 . . . . . n) if and only if secondary arc ( i , j ) is used in the path from the root to node p and Yi~ p = 0 otherwise.

L. Gouceia, 15. Janssen / European Journal of Operational Research 105 (1998) 552-568

2.1. Formulation DMF

555

I I 2 2 min E ( c i j X i j § c i j X i j ) (i . j)~A

subject to

E S=l ..... ,,, i :( i . j )~A

E ( ry + r y ) - E r,,-)=0, i :( i , j )~A i:( j , i )~A

E ( r i } ' + Y i ~ ' ) - - 1 , j = l . . . . . n, i :( i , j )~A

YiJP <_Xi~, ( i , j ) ~ A ; p = 1 . . . . . n,

Yi~t '<X~, ( i , j ) ~ Z ; p = 1 . . . . . n,

E (w.ry _<.. (i . j)~A

E r y - E i :( i , j )~A i:(j , i)~A

Yiji=O, ( i , j ) ~ A ; , = l , 2 ,

, 2 { 0 , 1 } ( i , j ) ~A, Xij , X i j E

p = 1 . . . . . n,

j , p = l . . . . . n; j ~ p ,

(Dl )

(D2)

j , p = l . . . . . n; j ~ p, (D3a)

(D3b)

(D4)

(D5)

(D6)

(D7)

(DS)

(D9)

(D10) Y/)P,Y/~PE{0,1}, ( i , j ) E A ; p = 1 . . . . . n.

Together with the binarycondifions on the variables, (D2) states that each node is in the solution. Constraints (D3a) and (D3b) are flow conservation constraints and together with the forcing constraints (D4) and (D5) they guarantee that the selected set of arcs is a connected solution. Constraints (DS) indicate that an arc diverging from a given node i is not used in the path to that same node. In connection with the binary constraints, constraints (D2), (D3a), (D3b), (D4), (D5) and (DS) model a directed MST problem, with two types of arcs for each pair ( i , j ) ~ A . Hop constraints are modelled by (D6) and finally, (D7) ensures that the primary arcs constitute a connected set containing the root node.

In the following let PL denote the LP relaxation of model P and let v ( P ) denote the cost of its optimal solution.

Notice that the flow conservation constraints for node 0

E ( ro ' l '+ romp)=1, p = l . . . . . ,,, (D3c) j:(O,j)EA

are redundant in DMF L. In fact, by summing up all constraints (D3a) for a given p, summing up constraints (D3b) for j = p and using (D8) we obtain the constraints (D3c) for the same p. Alternatively we might have included such constraints in the DMF model. Then, in a similar manner we could have shown that constraints (D8) would be redundant in the LP relaxation of such a model.

It is important to point out that constraints (D4) and (D5) for j = p are satisfied as equalities in DMF c. To see this, notice that (D4) and (D5) imply

556 L Gouceia, E. Janssen / European Jo,mlal of Operational Research 105 (1998) 552-568

Yi~ t' + _,,Y.2P_Xi~ + (i , j) ~A; p = I,. .. ,n.

Now, (D2) and (D3b) imply that the above constraints are satisfied as equalities for j = p, i.e.

Yi~ j + YU2j-- Xi ~ + X~, ( i , j ) ~Z .

The above equalities now show that Y~[~. < X~t.y for some i* if and only if YfiJ > X~.j for the same i ' . But the second inequality is in contradiction with (D5). In the same manner we can prove that we cannot have ri~ j < X~.

An undirected model (which uses edge variables instead of arc variables) was also presented in Janssen (1995). The LP relaxation of such a model is equivalent to the LP relaxation of the model given above (the proof of this result, which follows closely the proofs given in Magnanti and Wolsey, 1995 and Balakrishnan et al., 1994 for related problems is also given in Janssen, 1995). The undirected model uses less variables than the directed model, however it uses O(n 2- I AI) constraints which is a much larger number than the number of constraints O(n . I AI) involved in the directed model. The smaller number of constraints used by the directed model makes this model much more attractive to use in lower bounding schemes.

3. Alternative pr imary connectivity constraints derived from the directed tree polytope

It will be interesting to discuss several alternative sets of constraints which can be taken from the literature related to similar network design problems and which also model the primary connectivity requirements involved in the HPCTLMST problem. The strength of these constraints will be compared with the strength of the constraints (D7) which are used in our model. Contrary to constraints (D7), the alternative constraints which are discussed in this section involve only the arc design variables Xi~. The first set of such constraints is 'borrowed' from a directed Steiner tree formulation given in Khoury et ai. (1993). Consider, then, the following set of constraints

E I I Xki>__Xij, ( i , j ) ~ A ; i > O . (1) k:(k, i)~A

The above constraints guarantee that the arc ( i , j ) is primary only if there exists a primary arc (k,i) entering node i. Noticing that k should he different from j leads to the following lifted version of (1).

X~i> Xitj, ( i , j ) ~ A ; i > O . ( i ' ) k:(k,i)EA

We shall show next that constraints (1') are redundant in DMF L. Before showing this, let us start by considering a generalization of the inequalities (1') given by

xi~>_ ~ x~p, S _ { I . . . . . n}; ISl> l; p = l . . . . . n ; p ~ S , (2) iESC-{p} i~S ( i , j )~A (i ,p)~A

where S c = V - S. Notice that when ISI = 1 we have precisely constraints (1'). Notice that by summing up (D7) for j ~ S we obtain

~ y]., >_ ~ ~ yj~p>_ ~ yj~p, S__{1 . . . . . n } ; I S l > _ l ; p = l . . . . . n ; p ~ S . iES c j ~ S tI (D7) j ~ S i~SC-{0} j ~ S

( i , j )~A ( j , i )~A ( j ,p)~A

L. GouL'eia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568 557

Using (D4), (DS) and the fact that constraints (D4) are satisfied as equalities when j = p leads to

E E EYY>- E Ex],, iESC--{p} j ~ S iES e j ~ S deS dES

( i , j )~a ( i , j )~A (j ,p)~A ( j ,p)eA

so_{1 . . . . . n}; ISl> I; p = 1 . . . . . n; pf~S.

By transitivity we obtain (2) which means that these constraints (including (1')) are redundant in DMF L. Note that the set of primary arcs form a not necessarily spanning tree directed away from node 0. Therefore it

would be worth to consider constraints which are involved in formulations for the directed tree polytope. The following set of constraints is 'borrowed' from a directed tree formulation discussed in Goemans (1994):

E E Xi~>-- E Xilp, S _ V { 0 } ; p ~ S . (3) i~S c jES i~V

(i.j)sA ( i ,p )ea

These constraints are known to be involved in the description of the convex hull of the incidence vectors of directed trees rooted at node 0 when the underlying graph is series-parallel. Constraints (3) state that if a primary arc converges into node p then at least one primary arc should be included in each cut [Sr such that p ~ S. Constraints (3) are equivalent to constraints (2). To see this, notice that (2) can be rewritten as

]~ x}j>_ Y'. X,~, S___{1 . . . . . n};ISl>__2;p~S. (2') i eS c j e S - { p ) iES-{p}

( i , j )eA (i,p)EA

By adding ~ iEs c Xilp to each member of (2') we obtain (3). (i,p)EA

At this moment we do not know whether the set (3) (or (2)) is equivalent to (D7) in the sense that the two models DMF, and DMF with (D7) replaced by (2) produce the same LP value. However it was shown that (D7) dominates (3). This indicates that by using the compact set of constraints (D7) in DMF L we are implicitly verifying a huge set of alternative primary connectivity constraints which are known to be facet defining for a related problem which is contained in the HPCTLMST problem. Additionally there is another reason why we prefer to use the set (D7) instead of the exponential set defined by (3) or the weaker set defined by (1) or (l ') . The Lagrangean relaxation discussed in Section 5 dualizes the coupling constraints (D4) and (D5). The relaxed problem obtained in this way can be separated into two subproblems, one involving the X[/ variables, the other involving the Yi~ p variables. As shown in Section 5 both subproblems can be efficiently solved. On the other hand, if we had used instead any of the alternative sets of primary connectivity constraints we would have to dualize the inequalities in such a set because we do not know how to solve the resulting subproblem defined on the X[/ variables.

Our previous analysis indicates one way of deriving an extended and compact description of the convex hull of the incidence vectors of directed trees rooted at node 0 when the underlying graph is series-parallel. Goemans (1994) has presented such a description using the binary variables Xi/ which indicate whether arc (i,j) is in the optimal tree. Such a set of linear inequalities is as follows.

E Zii<_ 1, j = 1 . . . . . n, (DT1) i : ( i , j )eA

E E Xi/>- E Xip, S G V { 0 } ; p E S , (DT2) i eS r d eS i~V (i , j)~A (i,p)EA

Xu>O, ( i , j ) ~ A . (DT3)

558 L. Gouceia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568

By using the extra variables Yi~ indicating whether arc ( i , j ) is used in the path to node p, using the constraints (DT1) and (DT3) plus

E Yi p >- E YjPi, J,P = 1 . . . . . n; j 4: p, (DT4) i:(i,j)EA i:(j,i)EA

Yi p <- Yij, ( i , j ) E A; p = 1 . . . . . n; j 4= p, (DTSa)

Yi~ = XU, ( i ,J) ~ A, (DT5b)

Yi~ = O, ( i , j ) ~ A, (DTr)

YiP>_O, ( i , j ) ~ A ; p = l . . . . . n ; i q : p , (DT7)

and following the argument used to derive (3) from (D7), (D8) and (D4) we can show that the system (DT1), (DT2) and (DT3) is the projection into the subspace defined by the X U variables of the polytope defined by (DT1),(DT3) . . . . . (DT7). Therefore, together with Goemans result we have shown that the system defined by (DT1),(DT3) . . . . . (DT7) gives an extended and compact description of the polytope associated to directed trees rooted at node 0 when the underlying graph is series-parallel.

4. Simple hop constraints

Notice that in the integer formulation the number of primary (secondary) weight of a path can only be a multiple of w~ (w2). This implies that the following simple hop constraints are valid for the HPCTLMST problem.

E Y,l"<-tH/w,l, p = I . . . . . n , ( 4 ) (i,j)EA

Yi 2p <_ [ H / w 2 ] , p = 1 . . . . . n. (5) (i,j)EA

We shall denote by HDMF L the model DMF L augmented with (4) and (5). Results given by the models DMF L and HDMF L will be presented in Section 9.

The DMF L model for the largest problem which we shall be examining, one with 40 nodes, contains about 73 800 variables and about 82 400 constraints. Solving the LP relaxation of such large models will be extremely time consuming and will lead to impossible storage requirements for most of the machines available at the moment. Therefore we need alternative methods for computing an approximation of the corresponding LP bounds. Such an alternative method is given by Lagrangean relaxation which will be discussed in the next section. Additionally, we shall show that the theoretical best bound associated with this Lagrangean bound dominates the bound given by the cost of the optimal solution of HDMF L.

5. A Lagrangean relaxation scheme

By penalizing the forcing constraints (D4) and (D5) we derive a Lagrangean relaxation for the problem. We attach nonnegative Lagrangean multipliers K/~ > 0 (for ( i , j ) ~ A; p = 1 . . . . . n) and A p >_ 0 (for ( i , j ) ~ A; p = 1 . . . . . n) respectively to the forcing constraints (D4) and (D5) and put them in the objective function in the usual Lagrangean way. This leads to the following Lagrangean problem denoted by REL,,,~.

L. Gouceia, E. Janssen / European Journal o f Operational Research 105 (1998) 552-568 559

5.1. Formulation REL~a

min E (c]yXi~+ci~Xi2) + E ~kP(Yi}P-Xi~) + Y'. ~".A~(Yi 2p-X2) ( i . j ) ~ A ( i , j )EA p= I ( i , j )EA p= I

subject to

(D2), (D3a), (D3b), (Dr ) , (D7), (D8), (D9) and (D10).

The resulting problem can be separated into a problem in the X[j variables ((i,j)~ A; t = 1,2), denoted by RELx. ~ (involving the first summation term in the objective function and constraints (D2) and (D9)) and one in the Yi~ variables ((i,j) ~ A; t = 1,2; p = 1 . . . . . n), denoted by REL r , (involving the two remaining summa- tion terms in the objective function and constraints (D3a), (D3b), (D6),'(D7), (D8) and (D10)). The latter can be decomposed into n subproblems R E L r , . , one for each node p = 1 . . . . . n. The cost of the relaxed problem REL,,A is given by v(REL,, x) = v(RELx'I? + E~= Iv(RELr ). An approximation of the optimal multipliers can be obtained by standard subgradient opti'nfization (see Helmet al., 1974). A well-known result from Lagrangean theory (see Geoffrion, 1974) states that v(D) = maxl,,zo. A~olREL,,x > v(DMFL), where v(D) is the value of the Lagrangean dual problem. We shall show later that strict inequality holds for some instances.

The problem RELx. ' is an inspection problem for every value of j ( j = 1 . . . . . n). The optimal solution to this problem is obtained by taking for every node j ( j = 1 . . . . . n) the least cost arc, either primary or secondary, entering node j. For general arc costs, the optimal solution to each subproblem REL r, , may not be connected but since the values K~ and A~ ((i,j) ~ A; p = 1 . . . . . n) are nonnegative the optimal ~olutions to each of these subproblems will correspond to elementary paths (paths in which every node appears only once). Therefore, each REL r corresponds to the problem of finding the shortest hop constrained path from the root to node p subject to t~e constraint that the primary arcs constitute a (possibly empty) subpath starting at the root. As we shall show next, for a given p each subproblem RELr,,. ~, can be solved efficiently by dynamic programming.

For defining the dynamic programming equations we shall use the following values

SP~(i,q) = the cost of the shortest path from the root to node i with at most q primary arcs and no secondary arcs (i = 0 . . . . . n, q = 0 . . . . . [H/wl]);

SP2(i, q) = the cost of the shortest path from node i to node p with at most q secondary arcs and no primary arcs (i = 0 . . . . . n, q = 0 . . . . . [H / /w2] ) .

Then

v(RELr.~) = min ( rain [SP,( i ,q) + SP2(i,[(H-q.w,)/w2l)] } [i=o ..... n] lq=O ..... IH/,-I]]

The equality above follows from the fact that the optimal path from node 0 to node p is composed by two optimal paths, one from node 0 to a certain node i containing only primary arcs and the other from node i to node p containing only secondary arcs. Clearly, each subpath (primary or secondary) can be empty.

To compute SP~(i,q) and SP2(i, q) we shall use the following, well-known recursive equations:

SPl(i,0) = 0 if i = 0 and equal

SPl( i ,q) = min/SPi( i , q - 1),

to co otherwise,

min (SP~j,q 1) KP)} f o r ( i = 0 . . . . n,q=l, . ,tH/,v,l), [ j : ( j , i ) ~ A ] " ( - - "1- 1 " ""

SP2(/,0 ) = 0 if i = p and equal to ~ otherwise,

SP2(i,q ) = min(SP2(/, q - 1), min ( S P , ( j , q - 1 ) + A/~)} [ j : ( i , j ) ~ A ]

f o r ( i = o . . . . . . , q = l . . . . . [ H / w z J ) .

560 L. Gouceia, E. Janssen / European Journal of Operational Research 105 (1998)552-568

3

~o / " 12c

1 2

4

/ / t

t

Fig. l. Instance where relevant lagrangean costs rij 4 for primary arcs (bold lines) and All 4 for secondary arcs (dotted lines) are indicated for the path problem to node 4, RELr, . .

It turns out that due to tile presence of the hop constraints, each subproblem in the Yi~ p variables does not satisfy the integrality property. As an example consider an instance with n = 4 and H = 5, w I = l, w 2 = 3. We examine the subproblem RELy4" in the graph depicted in Fig. 1, where the costs Kii4 (referred to by bold lines) and A~s 4 (referred to by dotted lines) have the values indicated. For simplicity we assume that the other costs K~s 4 and A~./4 have such large values that the corresponding primary, respectively, secondary arcs will never be selected in the solution.

The solution to R E L y , ' is described by Yol~ = y~4 = 1 with cost 22.0. However, when we relax the integrality constraints in R E L r , ~, the optimal solution is now given by Y0~ = 1, Yt�89 ~ = Y ~ = 0 . 5 and y ~ = y24 = 0.5 with cost 20.0. Thus, the relaxed problem does not satisfy the integrality property which implies that the corresponding dual Lagrangean bound can be better than the corresponding L P relaxation lower bound, i.e. v ( D ) = maxl, , ~. o, A> 0lv(REL,,,0 > v(DMFL) for some instances. Moreover, by observing that each feasible solution for each subproblem REL r satisfies the simple hop constraints (4) and (5) (notice the range of variation of q when defining SP t and SP2~'xve can also show that v(D) >__ v(HDMFL). Again, it is not difficult to see that strict inequality holds for some instances (see Section 9).

6. A L a g r a n g e a n heurist ic

Following Balakrishnan and Altinkemer (1992) our Lagrangean heuristic is also based on the values of the Yi~ p variables given by each relaxed solution. Note that the paths associated to the optimal solutions of the n subproblems defined in the Yi~ p variables, already satisfy the generalized hop constraints and the primary connectivity constraints. This suggests that only slight modifications may be needed to obtain a feasible solution for the integer program DMF. We shall start by giving a brief description of the heuristic procedure. Afterwards we shall present a more formal pseudo-Pascal program.

In phase 1, we construct a graph containing all arcs used in the n directed paths corresponding to the optimal solutions of the n subproblems defined in the Yi7 variables. If a particular arc appears in both technologies, we choose technology 1 for that arc. The resulting network contains at least one directed path from the root node to every other node which satisfy the generalized hop constraints. In general such a network will not be feasible to I IPCTLMST because it might contain a node that has more than one incoming arc. However, it can always be transformed to a directed tree that satisfies the generalized hop constraints and primary connectivity constraints. Suppose there is a node p which has two or more incoming arcs. For such a node p, we find the directed path from node 0 to node p with minimal weight, where the weight of a path is defined as the sum of the weights of the arcs included in the path. Let i" be the last node before p in this path. Delete all arcs ( i , p ) for i q: i ". The resulting network still contains at least one directed path to every node satisfying the generalized hop

L. Gouveia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568 561

constraints, because the directed path with the smallest weight to node p has been maintained in the network. If we continue this process, until all nodes (except the root) have exactly one incoming arc, we obtain a tree that satisfies the generalized hop constraints.

Assume now that after this procedure is performed, the remaining incoming ,arc into a particular node p is secondary and assume also that there are primary arcs diverging from node p. Then, the primary connectivity constraints are not satisfied for this solution. To guarantee the primary connectivity requirements, each time we have a secondary arc converging into node p and a primary arc is diverging from the same node, all arcs of the directed path to node p with minimal weight to node p that are secondary, are replaced by primary arcs. The resulting network then is a feasible solution to the problem. Finally, we apply a simple local search method to improve the solutions obtained by the process described above. We shall describe next the heuristic by a pseudo-Pascal code.

Phase 1 For all arcs ( i , j ) do Set Xi~ = 0 and X/~ = 0 For all arcs ( i , j ) do For all p do If Y ,27 : l then set X~=~ 1

| p If 1 then set Xi'..j = 1 and set X/~ = 0.

Phase 2 _ _ t 2 _ _ For all nodes j for which there are at least two nodes k and q such that X~J - Xqj - 1 for some t l , t 2 = 1,2

do Determine a path P to j with lowest weight; let k* be the last node before j in this path For all k 4: k* such that X~j = 1 (t = 1,2) do Set X~j = 0 If X~..i = 1 and X],,, = 1 for a node m then For all arcs (k, l) in P such that X~t = 1 do Set X~t = 1 and set X~t = O.

Phase 3 Define S to be the set of nodes i for which there exist a node j and a technology t such that X[j = 1 For all nodes j not in S do For the node i and technology t such that X[j = 1 do set Xtj = 0 Define c~'j = min{i,rl{C~i: i ~ S, t = 1,2, such that installing a technology t arc ( i , j ) preserves hop and primary connectivity constraints}; set X].j = 1.

To illustrate the heuristic we shall use a small example. Consider the instance with 4 nodes and the root node with secondary cost matrix given by Table 1,

Let H = 5, w 1 = 1, w 2 = 3 and let the primary cost of an arc be equal to twice the corresponding secondary cost. In a particular iteration of the subgradient procedure let the solution to RELr~ ~ be given by Yo21 ~ = 1, yd~ = yl~2 = 1, Yo n = YI/3 = Y423 = 1 and Y01( = y,~4 = y ~ = Y3144 = 1. Then the non-tree network resulting after phase 1 of the heuristic is given by Fig. 2. Bold lines correspond to Xi~ = 1 and dotted lines correspond to X ~ = 1.

From the root node to node 3 there are two alternate (elementary) paths. The one with the lowest weight is the path with primary arcs (0,I), (1,2), (2,3). Therefore we delete secondary arc (4,3). Now, from the root node to node 4 there are also two alternate (elementary) paths. The one with the lowest weight is the path with

562 L. GouL'eia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568

Table 1

Secondary cost matrix for graphical example

1 2 3 4

0 I 0 20 22 30

1 - 10 14 20

2 - 10 11 3 - 5

I 2

Fig. 2. Network after phase 1.

3 4

I 2

Fig. 3. Network after phase 2, cost = 100.

$ 4

- !

/ /

/ \ ,; , /

I | i s

1 2

Fig. 4. Network after phase 3, cost = 61.

L. Gouveia, E. Janssen / European Journal of Operational Research 105 (1998)552-568 563

primary arcs (0,1), (1,4) so we delete primary arc (3,4). Note that in this case no additional alterations need to be made to guarantee the prima~-y connectivity requirements. The tree network resulting after phase 2 of the heuristic is given by Fig. 3. Note that if we had visited nodes 3 and 4 in phase 2 in reversed order, the resulting tree network would have been the same. It is not difficult to see that such a result is true in general.

Now, phase 3 defines S = {0,1,2}. Changing the primary arc (2,3) to the secondary arc (2,3) and the primary arc (1,4) to the secondary arc (2,4) leads to the solution given by Fig. 4 which turns out to be a better solution.

7. Problem reduct ion

There is a simple means of reducing the effort needed to solve our LP formulations and Lagrangean relaxation. We borrow a simple test from Gouveia (1996) which enables us to state that some arcs of a given type cannot appear in the optimal solution of a HPCTLMST instance.

If c~j > c~j for ( i , j ) ~ A; (0, j ) ~ A; t = 1 or 2 then arc ( i , j ) of type t can be removed and the same happens

with variables X/j and Yi~ ( P = 1 . . . . . n).

To check the validity of the test, notice that if the optimal solution contains such an arc ( i , j ) then we could replace it by arc (0, j) of the same technology, obtaining another feasible solution whose cost is not greater than the cost of the original solution. In the cases where for every arc the primary cost has the same proportion to secondary cost, the test enables us to remove an arc from the graph defining the instance. The reason for this is that if the test eliminates an arc of type 1 (2) then it also removes the same arc with type 2 (1).

8. Problem generat ion

For a given graph and secondary costs defined on the corresponding arcs we shall be considering several values of the weights w~ and w 2 and hop parameter H. For the relation between secondary and primary costs we shall use either proportional cost cases, i.e. c]j = K . c2j ( ( i , j ) ~ A; K > 1), where the constant K is the primary to secondary cost ratio or consider the case where for each arc, K is drawn from some uniform [a,b] distribution with a >_ 1. We consider two types of instances, one with euclidean costs and the other with random costs. For the euclidean costs we draw n points uniformly in a (0,100) • (0,100) grid and take the integer part of the distance between two points i and j as the costs for both arcs ( i , j ) and ( j , i ) . For the location of the root we use the (50,50) location. Random tests are constructed by drawing a random number from a uniform [0,100] distribution and taking the integer part as the costs for both arcs between any pair of nodes (including the root, in which case there is only one arc).

9. Computa t ional results

In this chapter we shall present a few results given by the LP relaxations DMF L and HDMF L and a large set of results given by the Lagrangean scheme, denoted by REL, which was described in Sections 5 and 6. The results of the LP relaxations have been produced using the GAMS 2.25 solver on a 486DX2-66 PC and results for the Lagrangean relaxation based schemes have been produced by a Pascal program executed on the same computer. To be able to take full advantage of the arc elimination test, our implementation only stores non-eliminated arcs. We describe next some implementational details of the subgradient optimization procedure. Lagrangean multipliers are initialized to zero. For updating the multipliers we use the following well-known formula for the step size = #r- (1.05 - UB - LB) / ( sum of squared subgradients), where UB means the best

564 L. Go,weia. E. Janssen / European Journal of Operational Research 105 (1998) 552-568

Table 2 Results for complete graphs with'n = 10, which compare the two LP relaxations with REL

Test K H )v I w 2 Opt DMF L HDMF L REL %Prim NEA

Random 3 5 I 3 210 180.8 184.5 210 50 44 (26) (42) (13)

2 4 1 3 190 178.6 178.6 190 40 (28) (31) (4)

2 4 1 2 160 151.7 - 160 60 (28) - (13)

Random 3 5 1 3 181 176.7 179.0 181 20 34 (95) (220) (9)

2 4 1 3 206 178.3 178.3 206 100 (30) (32) (6)

2 4 1 2 148 144.5 - 148 30 (56) - (5)

Euclidean 3 5 1 3 345 260.3 330.3 345 20 66 (13) (lO) (t)

2 4 1 3 294 286.8 294 294 20 (12) 14) (1)

2 4 1 2 237 236.5 - 237 0 (lO) - (1)

Euclidean 3 5 1 3 324 261.3 317.5 324 20 56 (20) (19) (3)

2 4 I 3 300 282.4 297.3 300 50 (22) (21) (2)

2 4 l 2 236 236 - 236 0 (15) - (1)

upper bound and LB is the value of the Lagrangean bound of the previous iteration. The parameter ~- is usually taken between 0.0 and 2.0. We found that the speed of convergence is substantially increased when using larger values. In our implementation the parameter is initialized to 8.0. If the lower bound does not improve for 5 consecutive iterations, the value of that parameter is halved. Every 50 iterations of the subgradient optimization procedure the parameter is set back to 4.0, to prevent the step size from becoming too small which, in turn, would lead to very small modifications in the values of the multipliers. Finally, we follow Beasley (1993) and use the factor 1.05 to prevent the step size from getting too small in case the bounds get close. Different settings for these parameters have been tried and our choice leads in general to the best bounds.

We shall start (see Table 2) with results given by the two LP relaxations and lower bounds given by REL for small problem sizes. We shall compare D M F L with H D M F L to see what is gained by adding the simple hop constraints to the original model. In all results that follow we use w, = 1. Note that this implies that adding primary simple hop constraints (4) to D M F L is useless. Therefore, for this set of tests, the model H D M F L only contains the secondary simple hop constraints. The number of el iminated arcs ( ' N E A ' ) , the percentage of arcs that are primary in the best found feasible solution ( '% Prim') , the cost of the optimal integer solution ( 'O p t ' ) and the CPU-times (written between brackets, in seconds and given just below the corresponding lower bounds) are also reported. The results are taken from complete graphs with 10 nodes plus the root node and a proportional primary to secondary cost ratio. Notice that the number of arcs in such instances is equal to 100. Two instances for each type of test and three settings of the problem parameters are tried for each problem instance. In the Lagrangean relaxation REL we use at most 500 iterations for the corresponding subgradient optimization procedure. However, for all instances tested fewer iterations were needed to obtain the optimum and the reported CPU-times indicate the time needed until the optimum is obtained.

L. Gouveia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568 5 6 5

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~ 0 0 0 0 0 0 0 0 0 0 0 o o o o o ~ o ~ 0 0 o o ~ 0 o o 0 ~ 0 o ~

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5 6 6 L. Gouueia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568

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L. Gouceia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568 567

Note that for H = 4 and w 2 = 2, v(HDMF L) cannot be better than v(DMF L) because constraints (5) are redundant in DMF L for these p~ameters. As expected, adding such constraints to DMF L turns out to be more efficient when both the remainder of the division of H by w 2 is larger and K is larger. Differences in CPU-times between the two kinds of tests are explained by the difference in the number of eliminated arcs. Notice that the number of eliminated arcs in the euclidean tests is bigger than the number of eliminated arcs in the random tests. REL finds the optimal solution very quickly for all of these small problems. The fact that the integrality property is not satisfied by the Lagrangean problem is indicated by the improvements on the LP bounds in some cases. These results also show that REL is a more efficient method for computing lower bounds than solving any of the LP models.

Next we shall present a set of results given by REL for larger instances. As a performance measure of the lower bounds we use the ratio (v(UB) - v(LB))/v(UB). 100%, denoted by 'Gap ' , where v(UB) and v(LB) are the best upper bound and the best lower bound (the last one rounded up to the next integer because the costs are integer) for the instance. The CPU-time in seconds needed to perform either the indicated number of iterations or to obtain the optimal solution is also given. To get an idea of the strength of the arc elimination test we report the number of eliminated arcs, denoted by 'NEA' . Note that for complete instances with 20, 30 and 40 nodes, the total number of original arcs is given respectively by 400, 900 and 1600. When the primary costs are not proportional to secondary costs we indicate the interval in which K is uniformly drawn for each arc. Note that in this case we need to distinguish between the number of primary and secondary eliminated arcs ( 'NEA1/2 ' ) . Due to the different sizes we decided to set a maximum of 2000 iterations of the subgradient optimization procedure for the 20 node tests, a maximum of 3000 iterations for the 30 node tests and a maximum of 4000 iterations for the 40 node tests. The choice for these limit values for the maximum number of iterations were based on a few computational experiments reported in Janssen (1995).

Tables 3 and 4 show no significant differences between the proportional and nonproportional cost cases. Differences in CPU-time between Euclidean and Random instances can be explained by a difference in the number of eliminated arcs. In fact, after preprocessing the euclidean tests become substantially smaller than the random tests. Additionally, the CPU-time gets bigger for the instances with bigger values of H and smaller values of w 2 because the number of iterations in the dynamic programming scheme also increases. Notice also that it becomes more advantageous to use primary arcs for larger values of w 2 and smaller values of K. This is confirmed by the value of % Prim.

These results show that the dual Lagrangean bound associated to REL is quite strong and the reported gaps are often close to zero. The Euclidean tests seem to be easier to solve than the corresponding Random tests but this might be explained by the number of eliminated arcs in each type of instance. The disadvantage of the proposed method is the time needed for obtaining the reported bounds. However, considering the large investments involved with the building of telecommunication networks, the reported CPU-times seem to justify the reported gaps.

10. Conclusion

In this paper we have introduced the hop and primary connectivity constrained two level minimal spanning tree problem which is related to the design of a centralized network with reliability requirements and two cable technologies are available. Since the problem is NP-Hard we have formulated it as a directed multicommodity flow model and derived a Lagrangean relaxation from such a model. We have derived a Lagrangean heuristic to provide upper bounds on the costs of the optimal solutions. The reported computational results for instances with up to 40 nodes indicate that for many of the cases tested the gaps are often close to zero. One side effect of our analysis of primary connectivity constraints for a directed multicommodity flow model for the problem was the derivation of an extended and compact description of the convex hull of incidence vectors of directed trees rooted at a given node when the underlying graph is series-parallel.

568 L. Go,weia, E. Janssen / European Journal of Operational Research 105 (1998) 552-568

A c k n o w l e d g e m e n t s

Th i s r e sea rch was par t ia l ly suppor t ed by Pro jec t No. P B I C / C / M A T / 2 1 3 0 / 9 5 . T h e c o m m e n t s o f the

re fe rees are grea t ly apprec ia ted .

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