ELSEVIER European Journal of Operational Research 94 (1996) 582-593

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Theory and Methodology

Tight linear programming relaxations of uncapacitated p-hub median problems

Darko Skorin-Kapov a’ * , Jadranka Skorin-Kapov a, Morton O’Kelly b a Center for Information Systems Managemeni, WA Harrimun School for Management and Policy. State University of New York,

Stony Brook, NY I1 794-3775, USA b Department of Geography, The Ohio State University, 103 Bricker hall, 190 North Oval Mall, Columbus, OH 43210, USA

Received 24 May 1994; revised 1 March 1995

Abstract

The problem of locating hub facilities and allocating non-hub nodes to those hubs arises frequently in the design of communication networks, airline passenger fiow and parcel delivery networks. In this paper we consider uncapacitated multiple and single allocation p-hub median problems. We develop new mixed O/l linear formulations with tight linear programming relaxations. The approach is tested on a well known and heavily used benchmark data set of real-world problems with resulting LP relaxations ranging from 10010 to 391 250 variables and from 2 101 to 3 1901 constraints, which proved to be difficult linear programs. Yet, this approach proved to be very effective: in almost all instances the linear programming solution was integer. In cases with fractional solutions, the integrality was achieved by adding a small partial set of integrality constraints. Therefore, we extended the range of optimally solvable instances of these NP-hard hub location problems, which have defied researchers for the last ten years. As an additional result for the single allocation case we were able to establish optimality of all heuristic solutions obtained via tabu search algorithm from a previous study. For the more difficult single allocation p-hub median problem we also used the best known heuristic solution as a guidance in adding integrality constraints. This novel linkage between optimal and heuristic solutions has a potential impact in a number of other problem settings, where efficient heuristic solutions exist and are probably, but not provably optimal.

Keywords: Hub location; Linear programming; Integer programming; Heuristic solutions; Tabu search

1. Introduction bell (1994a). He presented mixed O/l linear pro-

The problem of locating hub facilities arises fre- quently in the design of communication networks, as well as in airline passenger flow and parcel delivery networks. In this study we concentrate on uncapaci- tated multiple and single allocation p-hub median problems, which were recently addressed by Camp-

gramming formulations for those problems. Integral- ity restrictions imposed on a subset of variables, coupled with the large size of formulations (for an n-node network, the number of variables is 0(n4)> restricts the suitability of those formulations to small instances. In order to increase optimal solvability of hub location problems, tighter LP relaxations are needed. Since no computational results were given in

* Corresponding author. E-mail: dskorin@edge.har.sunysb.edu; Fax: (5 16) 632-9412.

Campbell (1994a), we first tested some of Campbell’s models. LP relaxations of those models resulted with

0377-2217/96/$1.5.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00100-X

D. Skorin-Kapov et al./ European Journal of Operational Research 94 (1996) 582-593 583

Fig. 1. List of cities in the CAB data set: (1) Atlanta; (2) Baltimore; (3) Boston; (4) Chicago; (5) Cincinnati; (6) Cleveland; (7) Dallas-F.W.; (8) Denver; (9) Detroit; (10) Houston; (11) Kansas City; (12) Los Angeles; (13) Memphis; (14) Miami; (15) Minneapolis; (16) New Orleans; (17) New York; (18) Philadel- phia; (19) Phoenix; (20) Pittsburgh; (21) St. Louis; (22) San Francisco; (23) Seattle; (24) Tampa; (25) Washington, DC.

highly fractional solutions. In this paper we propose new mixed O/l linear formulations whose linear programming relaxations often provide integral solu- tions. As compared to Campbell’s models, it is inter- esting that the tight LP relaxations are achieved without increasing the number of variables, and for the multiple allocation version of the problem the number of constraints is actually decreased. This is in contrast to a number of IP formulations for unca- pacitated facility location problems, where a tighter LP is achieved by “decomposing” the constraint that clients can not be served from a location, unless a facility is placed in that location. (See for example Nemhauser and Wolsey, 1988, p. 15.)

The LP formulations are tested on a well known data set from the literature (for a recent survey see Campbell, 1994b). The data set originates from the Civil Aeronautics Board (CAB) and consists of 25 interacting nodes (see Fig. l), for which flows and distances between every two nodes are given. Sub- sets of these data are taken so as to generate 10 X 10, 15 X 15, 20 X 20 and 25 X 25 interaction systems. O’Kelly (1987) formulated the single allocation p- hub median problem (referred to as the hub location problem) as a quadratic O/l program with a noncon- vex objective function. A number of heuristic algo- rithms to solve this problem was proposed, for exam- ple by O’Kelly (19871, Klincewicz (1991, 1992) and Skorin-Kapov and Skorin-Kapov (1994). The best

known solutions for the above problems were ob- tained by the tabu search heuristic of Skorin-Kapov and Skorin-Kapov (19941, in which the allocations of non-hub nodes were based jointly on distances as well as on flows between the nodes.

For the above data set, the formulations proposed in this study result in LP relaxations ranging from 10010 to 391250 variables and from 2 101 to 31901 constraints, which proved to be difficult linear pro- grams. Nevertheless, the formulations turn out to be extremely useful: in almost all instances (115 out of 120) the obtained linear programming solution was integer. Where this was not the case, the LP objec- tive function value for the multiple allocation (resp., single allocation) case was less than 0.1% (resp., 1%) below the optimal integer objective function value. By exploiting the LP solution, and due to excellent lower bounds from the LP relaxation, the integrality can be easily achieved by adding a partial set of integrality constraints. For the multiple alloca- tion p-hub median problem there are n integrality constraints (i.e., n O/l variables), but for the data set we tested, adding only one of those constraints was sufficient to obtain an optimal integral solution. For the single allocation p-hub median problem there are n2 O/l variables, and therefore it is more difficult to decide which integrality constraints should be added to obtain an optimal integral solution. In this case the best known heuristic solutions for the given data set (obtained via tabu search) were also used as a guidance in adding integrality constraints. As an additional result we were able to establish the optimality of all heuristic solutions obtained via tabu search algorithm from Skorin-Kapov and Skorin- Kapov (1994). In sum, the main contributions of this paper include: (1) extension of the range of opti- mally solvable examples of hub location problems of real-world instances that were subject of numerous studies for more than ten years; (2) confirmation of solution quality of a known tabu search heuristic, so that with a high level of confidence one can use that heuristic for solving even larger problems; and (3) development of a novel linkage between optimal and heuristic solutions by using an existing heuristic solution as a guidance in developing strategy to obtain an optimal solution.

The plan of the paper follows. In Section 2 we first state the p-hub median problem and present the

584 D. Slwrin-Kapov et al./ European Journal of Operational Research 94 (1996) 582-593

IP formulation due to Campbell (1994a). We next propose new formulations in order to get tighter LP relaxation for the multiple, as well as the single allocation versions of the problem. In Section 3 we present the computational results, and in Section 4 we conclude with some directions for further re- search.

2. The p-hub median problem

The p-hub median problem can be described as follows. Suppose there are n nodes that should inter- act, and p of those should be designated as hubs. The objective is to facilitate interactions between nodes of the network via a set of hubs. The hubs are assumed to be fully interconnected, but the non-hub nodes can interact only via hubs. The hubs are uncapacitated, i.e., there is no restriction on the number of nodes allocated to a given hub. If there is no restriction on the number of hubs to which a non-hub node can be allocated, we have the multiple allocation version of the problem. A connectivity protocol in which each non-hub node is allocated to exactly one hub corresponds to the single allocation p-hub median problem.

2.1. Multiple allocation p-hub median problem

We first consider a less restricted problem, i.e. the multiple allocation p-hub median problem. Using the notation from Campbell (1994a). let us define the following variables: xijkm is the fraction of flow from location (origin) i to location (destination) j, routed via hubs at locations k and m in that order; yk = 1 if location k is a hub, and 0 otherwise. The input data are given as: n, the number of locations; p, the required number of hubs to be open; Wij, the flow from location i to location j; cij, the cost per unit of flow from location i to location j (it is proportional to the distance); OL 5 1 is the discount on the unit cost of flow between hubs. The cost per unit of flow between origin i and destination j, routed via hubs k and m in that order, is given by ‘ijkm =Cik+(YCkm + cmj. We assume that cii = 0, i= 1 , . . . ,n, so the formula for cijkm remains valid when i and/or j is a hub. Campbell (1994a) formu- lated the p-hub median ( p-HM) problem as follows.

p_HM

i=I j=l k=l m-l

subject to

k- I n n

C Cxijkm=l, i=l,..., n,j=l,..., n, k-1 m=l

( 1.2)

xijkm 5 yk, i= l,..., n, j= l,..., n, k= l,..., n,

m= I,...,n, (1.3) xijkm r; y,,,, i= l,..., n, j= I,..., n, k= l,..., n,

m= l,...,n, (1.4)

yk~ {O,l}, k= l,..., n,

Xijkm 2 0, i= 1 ,..., n, j= l,..., n, k= l,..., n,

m= I,...,n. The objective is to minimize the overall transporta- tion cost subject to: having exactly p hubs (con- straints 1); the flow between every origin-destination (o-d) pair (i, j) should be routed via some hub pair (constraints 2); and flows can be routed only via locations that are hubs (constraints 3 and 4). Vari- ables y, serving as hub indicators, are restricted to be 0 or 1, and flow variables x are nonnegative. Due to constraints (2), it is clear that x variables can not have values bigger than 1. Problem p-HM is a very large mixed O/l linear problem (with n f n4 vari- ables, and 1 + n2 + 2n4 constraints). Campbell (1994a, p. 390) states “In the absence of capacity constraints on the links, an optimal solution will have all xijkm equal to zero or one since the total flow for each o-d pair should be routed via the least costly pair”. Yet, due to its size, the integrality of y variables makes the problem very difficult to solve. Relaxing the integrality, however, results in a highly fractional solution (see Section 3).

An intuitive explanation for obtaining fractional solutions when relaxing the integrality of y variables is that the constraints (3) and (4) are not “strong enough” with respect to hub locations. Namely, since there are no fixed costs for opening the hubs, relaxing integrality will create lots of “partial” hubs,

D. Skorin-Kapov et al./ European Journal of Operational Research 94 (1996) 582-593 585

depending on the cheapest routes indicated via xijkm variables. We propose a modification to Campbell’s model p-HM as follows. Replace constraints (3) and (4) by:

i Xijkm s yk 9 i= 1 ,..., n, j= l,..., n, m=l

k= l,...,n,

i ‘ijkm symy i= 1 ,...,n, j= 1 ,.**, n, k= 1

m=l ,***, n, in order to obtain the following multiple allocation p-hub median problem:

p-I-In&MA

subject to

k Yk=P’

k-1

k kxijkm=l, i=l,..., n,j=l,..., n,

(2.1)

k-lm-I _ (2.2)

2 xijkm5yk, i=l,..., n, j=l,..., n, m=l

k= l,...,n, (2.3)

k Xiikm sYm~ i=l ,...,n, j= 1 ,*.*, n, k-l .

m=l ,*.., n. (24 y,E {O,l}, k= l,..., n,

xijkm 20, i= l,..., n, j= l,..., n, k= l,..., n,

m=l ,*.*, n. Note that p-HM-MA is equivalent to p-HM, since for a given i, j, and k (respectively, i, j, and ml, only the cheapest route i-k-m-j will “survive”. However, the linear relaxation of p-HM-MA is tighter than the linear relaxation of p-HM. This follows because every nonnegative solution satisfy- ing (2.1)-(2.4) satisfies (l-l)-(1.41, but not vice versa. Intuitively, we expect that y variables will have values closer to integral values. Our computa-

tional results confirmed it, and in almost all cases the LP relaxation of p-HM-MA model provided integral solutions. For the instances with non-integral LP solutions, the LP relaxation resulted with objective function value less than 0.1% below the optimal objective function value (see Tables 1 and 2 in Section 3). Moreover, when compared to Campbell’s model (Campbell, 1994a1, the constraint set has been reduced by 2n3(n - 1) constraints.

2.2. Single allocation p-hub median problem

Solutions to the LP relaxation of p-HM-MA ex- hibit multiple allocation property. Namely, for a given o-d pair (i, j), it might be best to use the link from i to the hub k, but for the o-d pair (i, I>, it might be best to use the link from i to some hub other than k. In some applications it might be eco- nomically justified to restrict non-hub nodes to be connected to exactly one hub. Such a connectivity protocol is very sparing in its use of linkages, and has advantages in terms of the amalgamation of flows into efficient bundles. Campbell (1994a, p. 393) proposed the following mixed O/l linear for- mulation for the single allocation p-hub median problem:

i=l j=l k-l m-l

subject to

2 Yk=P’ (3-l) k= I

n n C C xijkm= 1, i= I,..., n, j= I,..., n,

k=l m=l

(3.2)

zik<yk, i= l,..., n; k= l,..., n, (3.3)

j=l m=l j= I

i= 1 ,..., n, k= l,..., n,

yk~{O,l}, k= l,..., n,

zik~{O,l}, i= l,..., n; k= l,..., n,

586 D. Skorin-Kapov et al./European Journal of Operational Research 94 (1996) 582-593

Xijkm 2 0, i=l ,...,n; j= 1 ,. . .,n; k= l,.. . ,n;

m=l ,...,n,

where variable zik = 1 if location i is allocated to hub at location k, and 0 otherwise. The constraint set (3.4) enforces a single allocation requirement: for a given allocation of node i to hub k, the total flow between i and all other nodes j has to be equal to the total flow from i to all other nodes j via the link (i, k). Therefore, i can not be allocated to another hub, if it is already allocated to hub k. Campbell also proposes a slightly different formulation (to be re- ferred to as p-HM-IL’) by replacing constraints (3.4) with

zik+zj,,,-2xijkmz0, i=l)...) n,j=l,..., n,

k=l ,..., n, m= l,..., n.

Linear relaxations of Campbell’s formulations p- HM-1L and p-HM-lL’, however, are not tight, and lead to fractional solutions with objective function values significantly below the optimal objective functions values (see Section 3). In the sequel, we propose a new formulation for the single allocation p-hub median problem by modifying our model for the multiple allocation version of the p-hub median problem ( p-HM-MA).

The formulation p-HM-MA is modified by mak- ing the allocation choice of an origin node i indepen- dent of a destination node, and vice versa. To that end, let us use the “allocation” variables zik. With the assumption that a hub is allocated to itself, we candenote zkk=yk, k= l,..., n.

The following mixed O/l formulation could then be used for the single allocation p-hub median prob- lem:

p-HM-SA’

i-1 j=l k-1 m-1

subject to

i Zkk =Pv k- 1

(4.1)

k zik= 1, i= l,..., n, k=I

(4.2)

k xijkmIzkk, i= l,..., n, j= l,..., n, m-l

k= l,...,n, (4.3)

2 xijkmIzmm, i= l,..., n, j= l,..., n, k= I

m= l,...,n, ( 4.4)

i xijkm=zik, i= l,..., n, j= l,..., n, WI=1

k= l,...,n, (4.5)

2 xijkm=zjm, i= l,..., n, j= l,..., n, k= I

m= ,...,n, 1 t 4.6)

zikE {O,l}, i= l,..., n, k= l,..., n,

Xijkm 10, i= 1 ,...,n, j= 1 ,..., n, k= l,..., n,

m= l,...,n.

The changes from the p-HM-MA model occur in constraints (4.2), (4.5), and (4.6). Constraints (4.2) now state that each node has to be allocated to exactly one hub, constraints (4.5) assure that for every destination j, the sum C, xijkm (i.e., the total flow from origin i to destination j routed via all paths using link i-k) will be nonzero only if location i is allocated to hub k (independently of a destina- tion). Similarly, constraints (4.6) assure that for ev- ery origin i and every hub k, a flow through the path i-k-m-j is feasible only if j is allocated to hub m (independently of an origin).

In the sequel we will add further insight into the formulation p-HM-SA! by using quadratic program- ming formulation to redefine the meaning of x vari- ables. O’Kelly (1987) formulated the single alloca- tion p-hub median problem as a quadratic O/l problem. Due to the nonconvexity of the objective function, the continuous relaxation of the problem does not provide a useful approach towards its solv- ability. The problem has, therefore, been approached by a number of heuristic methods. For the sake of completeness we state the quadratic O/l formulation for the single allocation p-hub median problem.

D. Skorin-Kapov et al./European Journal of Operational Research 94 (19%) 582-593 587

p-HM-1Q n n n n

Inin c c c c wij(cik Zik + a Ckm ZikZjm i= I j- 1 k= 1 m= I‘

+ ‘jm Zjm)

subject to

i Zkk =Pv k- I

2 zik= 1, i= l,...,n;

(5.1)

(5.2) k= I

Zik s zkk 3 i=l ,..., n, k= l,..., n, (5.3)

zik~{O,l}, i= l,... n, k= l,..., n.

The above problem can be linearized by introducing the variables xijkm = zikzj,,,. It can be then shown that the resulting linearization is equivalent to p- HM-SA’. Frieze and Yadegar (1983) proposed such type of linearization in the context of the Quadratic Assignment Problem (QAP). Recall that an instance of the QAP of dimension n can be characterized as follows: Given n objects and n locations, find the minimal cost assignment of objects to locations by jointly taking into account flows among objects and distances among locations. However, for the QAP the above linearization is not as efficient as it is for the uncapacitated single allocation p-hub median problem we are addressing. This is due to the fact that the QAP is a “capacitated” location problem: at each location at most one object could be placed.

We next show that, without loss of generality, p-HM-SA’ formulation can be further simplified to obtain the following mixed O/l linear program:

p-HM-SA

subject to

t Zkk=P, (6-l) k-l

5 zik= 1, i= l,..., n, (6.2) k- I

Zik s Zkk 9 i= 1 ,..., n, k= l,..., n, (6.3/4)

i xijkm=zik, i=l,..., n,j=l,..., n, m=l

k= l,...,n, (6.5)

k xijkm=zj,, i= l,..., n, j= l,..., n, k= 1

m= l,...,n, (6.6)

zik E (OJ}, i= l,..., n, k= l,..., n,

xijkmrO, i= l,..., n, j= l,..., n, k= l,..., n,

m=l ,*.*, n.

By using the definition of z variables from con- straints (6.5) and (6.61, we replaced 2n3 constraints (4.3) and (4.4) from p-HM-SA by n2 constraints (6.3/4). The resulting formulation p-HM-SA has n2 O/l variables, n4 continuous variables, and 1 + n + n2 + 2n3 linear constraints. The linear relaxation of p-HM-SA is tight, as supported by our computational results: in almost all cases the LP solution was integral. For the instances with non-integral LP solu- tions, the LP relaxation resulted with objective func- tion value less than 1% below the optimal objective function value (see Tables 3 and 4 in Section 3).

2.3. Achieving integrality of solutions

In previous sections we formulated the multiple (respectively, single) allocation p-hub median prob- lem p-HM-MA (respectively, p-HM-SA). In each case we first solved the LP relaxation, and in 97% of the cases for the multiple allocation (resp., 95% of the cases for the single allocation) LP solutions were integral. When this was not the case we needed to impose some integrality restrictions. For the multiple allocation p-hub median problem there are n O/l variables (“hub indicators”). Based on LP solutions, we start adding integrality constraints until an opti- mal integral solution is achieved. Tight lower bounds from LP significantly shortened the enumeration tree. Actually, for the data used in this study, adding a single integrality constraint was sufficient to obtain an optimal integral solution for the multiple alloca- tion version of the problem.

For the single allocation p-hub median problem p-HM-SA, there are n2 O/l variables (“allocation

588 D. Skorin-Kapov et al./European Journal of Operational Research 94 (1996) 582-593

variables”). When the LP relaxation attains a frac- tional solution, it is more difficult than in the multi- ple allocation case to decide on branching variables (i.e. added integrality restrictions) when performing the implicit enumeration. For many NP-hard prob- lems there are effective, fast heuristics available, but it is often impossible to prove the optimality of those heuristic solutions. The challenge in such case might be in finding ways to use a known heuristic solution (probably but not provably optimal) and its objective function value, as sources of information when searching for optimal solution. In O’Kelly et al. (199.5) a novel approach of using a known heuristic solution (i.e., an upper bound) to derive a lower bound was proposed in the context of the single allocation p-hub median problem. In this paper, we use the best known heuristic solution in conjunction with the solution to the LP relaxation as a guidance for branching strategy towards achieving optimal solution for p-HM-SA model. We used the follow- ing heuristic rule to determine the branching vari- able: branch on the fractional hub variable zjj with the biggest sum of differences between values of variables zij, i= l,..., n of heuristic and LP solu- tions.

3. Computational results

The computational results were carried on using CAB data set. The calculations were performed on a Sun SPARC station 2, using the dual simplex algo- rithm from CPLEX (1993) for solving linear pro- grams. AMPL modeling language was used to facili- tate entries of formulations (Fourer et al., 1993). When needed to guide the branching strategy for the single allocation p-hub median problem, the best known heuristic solutions for the single allocation p-hub median problem obtained by the tabu search algorithm of Skorin-Kapov and Skorin-Kapov (1992) were used. The data consists of sets of 10, 15, 20, and 25 nodes, with 2, 3, and 4 hubs, and with the parameter 01 E {0.2,0.4,0.6,0.8,1.0) (60 problems overall). In general, the LP relaxations considered in this paper are difficult linear problems. Specifically, the LP relaxation of the problem with 4 hubs and 25 nodes translates to an LP with 391250 variables, 31901 constraints and 800000 + nonzero constraint

coefficients. This proved to be an extremely difficult linear program. We solved it using the dual simplex algorithm (from CPLEX 3.0), which is well suited for problems with little right hand side variability and significant variability in the cost coefficients. The time needed to solve instances of this linear program was 13- 15 hours of CPU time.

We first present more detailed results for the multiple allocation p-hub median problem, followed by the results for the single allocation case.

3.1. Multiple allocation p-hub median problem

Campbell’s formulation p-HM was applied to the data instance with 10 nodes, 2 hubs and (Y = 1.0. The solution was achieved after 55 minutes of CPU time, with hubs at nodes 7 and 9, and with the objective function value 721.20. The linear program- ming relaxation of p-HM gave the objective function value of 671.84 within few minutes of CPU time, with highly fractional solution. For example, the values of y variables were:

y, = 0.0952381 yz = 0.095238 1

y, = 0.0952381 y4 = 0.428571

ys = 0.142857 y, = 0.142857

y, = 0.285714 ys = 0.285714 yg = 0.142857 y,, = 0.285714

The linear programming relaxation of p-HM-MA formulation proposed in this study obtained integral solutions for 58 out of 60 problems considered in this study. The CPU time was dimension dependent and ranged from a minute (for 10 dimensional in- stances) to 3-4 hours (for 25 dimensional instances). The results are displayed in Table 1. Table 2 displays the problem instances for which LP relaxation did not obtain integral solution. In both cases the addi- tion of only one integrality constraint was sufficient to reach optimal integral solution. The relative gap between LP and IP objective function values was less than 0.1%.

3.2. Single allocation p-hub median problem

The LP relaxation of Campbell’s (1994) model p-HM-IL (respectively, p-HM- IL’) proposed for the single allocation p-hub median problem was applied

D. Skorin-Kapov et al./ European Journal of Operational Research 94 (1996) 582-593 589

Table 1 Multiple allocation p-hub median problem: Results from the LP relaxation of the p-HM-MA model

n P 0

0.2 0.4 0.6 0.8 1

10 2

3

4

15 2

3

4

20 2

3

A

25 2

3

4

Lpobj

(hubs) 612.12 662.11 698.69 713.55 721.20

7,9 7,9 7,9 7.9 739 487.26 543.73 586.47 625.48 654.35

4,697 4,677 4,697 4,677 4,697 389.04 466.53 530.26 589.36 632.18

33496.7 3,4,6,7 394,677 3,4,6,7 274,677

970.65 1033.86 1091.91 1108.79 1114.54 4, 12 4.12 4,12 4,7 4.7 783.80 864.58 941.21 1012.04 1039.39 4,7,12 4,7,12 4,7,12 4,7,12 1,4,7 626.33 738.52 841.78 927.42 989.24 4,7,12,14 4.7.12.14 4.7.12.14 1.4,7,12 1.4.7.8

972.25 1013.36 1046.89 1075.30 1090.63 4,17 4, 17 4, 17 11,17 11,lS 712.09 803.81 884.64 948.41 975.53 4,12,17 4,12,17 4,12,17 4.7.17 4,7,17 568.50 694.56 788.59 870.08 934.08 4,12, 16,17 1,4,12,17 4,7,12,17 4,7,12,17 4,16,17,19

996.02 1072.49 1137.08 1180.02 1206.62 12.20 12,20 12,20 12,20 12.20 752.91 859.64 949.23 1019.64’ 1061.38’ 12.17.21 4,12,17 4,12,17 non-integer non-integer 618.48 754.49 866.44 951.75 1006.66 4,12,17,24 4,12,17,24 1.4.12.17 1,4,12,17 1,4,12,17

‘LPobj’ indicates the objective function value of the linear programming relaxation of the p-HM-MA model; ‘hubs’ indicate hub locations whenever the LP solution is integer, otherwise it is stated that the solution is not integer.

to the data instance with 10 nodes, 2 hubs and OL = 1 .O, and obtained highly fractional solutions with the objective function value 659.85 (respectively, 650.68), while the optimal objective function value for this problem is 835.81.

The linear programming relaxation of p-HM-SA

non-integral LP solutions, the relative gap between LP and IP objective function values was less than 1%. The results are displayed in Table 3. CPU time was dimension dependent: around a minute for it = 10, few minutes for n = 15, 2-3 hours for n = 20, and 13- 15 hours for n = 25. The integral solutions

formulation proposed in this study obtained integral were identical to the heuristic solutions obtained by solutions in 57 out of 60 cases. For the cases with the tabu search method of Skorin-Kapov and

Table 2 Multiple allocation p-hub median problem: Achieving integrality by adding a partial set of integrality constraints to the LP relaxation

n P CL Lpobj Added integrality constraints Ipobj Hubs “obj - Lpobj

0.01 * LPObj

25 3 0.8 1019.64 y‘$ E IO, 1) 1020.04 4,12,17 0.04 25 3 1.0 1061.38 YaEIO.11 1062.14 12.18.21 0.07

590 D. Skorin-Kapov et al./European Jound of Operational Research 94 (1996) 582-593

Table 3 Single allocation p-hub median problem: Results from the LP relaxation of the p-HM-SA model

n

IO

15

20

25

P

2

3

4

2

3

4

2

3

4

2

3

4

Lpobj

(hubs)

Q

0.2

615.99 7,9 491.93

4,677 395.13 3,4,6,7

981.28 4.12 799.97 4,7,12 639.77

4,7,12,14

979.09 4,17 724.54 4,12,17 577.62 4,12,16,17

1000.91 12.20 767.35 4,12,17 629.63 4,12,17,24

0.4 0.6 0.8 I

674.3 1 732.63 790.94 835.81 7,9 799 7,9 497 567.91 643.89 716.98 776.68

4,6,7 4,6,7 4,7,9 4,7,9 493.79 577.83 661.41 736.26 4,6,7,8 4v6.7.8 4,7,8,9 1.4.7.9

1062.63 1143.97 1190.77 1221.92 4.12 4.12 4,11 4,ll 905.10 1009.93 1099.51 1167.23 * 4,7,12 4,7,12 4.798 non-integer 779.71 910.21 1026.52 1118.23

4.7.12.14 1,4,7,12 1,4,7,8 l,4,7,8

1042.57 1106.04 1169.52 1210.08 4.17 4, 17 4.17 4,20 847.77 970.99 1086.07 * 1156.07 4.12.17 4,12,17 non-integer 4,11,20 727.10 869.16 1008.49 1107.92 * 1,4,12,17 1,4,12,17 1.4.8.17 non-integer

1101.63 1201.21 1294.08 1359.19 12.29 12,20 12.20 8,20 901.70 1033.56 1158.83 1256.63 4,12, 18 2,4,12 2.4, 12 4.8.20 787.5 1 939.21 1087.66 1211.23 1.4.12.17 1,4,12,17 1,4,12,18 4.7.8.20

‘LPobj’ indicates the objective function value of the linear programming relaxation of the p-HM-SA model; ‘hubs’ indicate hub locations whenever the LP solution is integer, otherwise it is stated that the solution is not integer.

Skorin-Kapov (1994) when applied to the quadratic solution as a guidance in adding integrality con- formulation p-HM-1Q. Table 4 displays the three s&tints, we will discuss a couple of cases. cases in which the LP solution was not integral. In In the case n = 15, p = 3, (Y = 1.0, the objective order to illustrate the use of the best known heuristic function value of the LP relaxation of p-HM-SA was

Table 4 Single allocation p-hub median problem: Achieving integrality by adding a partial set of integrality constraints to the LP relaxation

n P ci Lpobj

15 3 1.0 1167.23 20 4 1.0 1107.92

20 3 0.8 1086.07

Added constraints

211.1, E (0, 11 211.11 EW.1) z2lJ,2lJ E IO, 1) 24.4 E IO, 1) 217.17 E KJlI ZaaE{O,l)

Ipobj Hubs

1168.68 4,7,8 1111.01 4.8.13.20

1091.05 4,8,17

lPtij - LPObj

0.01 . LPObj

0.124 0.279

0.458

591 D. Skorin-Kapov et al./ European Journal of Operational Research 94 (1996) 582-593

Table 5 Nonzero values of allocation variables in the p-HM-SA model with n = 15, p = 3, a = I.0

Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0.5 0.5 2 1 3 I 4 1 5 1 6 1 7 0.5 0.5 8 0.5 0.5 9 1

10 0.5 0.5 11 0.5 0.5 12 0.5 0.5 13 0.5 0.5 14 0.5 0.5 15 1

1167.23, with nonzero values of “allocation vari- ables” as shown in Table 5. The corresponding tabu search (TS) solution has the objective function value 1168.68, with the following solution (hub: alloca- tion)

4: 1,2,3,4,5,6,9,11,13,14,15; 7: 7,lO; 8: 8,12.

Note that location 11 is not a hub in the TS solu- tions, and yet in the LP solution, it has almost 50% of non-hub nodes partially allocated to it. In view of this difference, it seems that adding a constraint z,,,,i E (0, l} to the LP relaxation of p-HM-SA would provide a “strong cut”. In fact, that was enough to obtain an optimal integral solution, and the same objective function value as in the TS solution.

In the case with n = 20, p = 4, CL = 1.0, the objective function value of the LP relaxation of p-HM-SA was 1107.92, with nonzero values for the

LP (1107.92)

z1,,*1 =O (mo*)’ ‘* ,,,, 1 = l(llO9.09)

%o,#=o (1115.54) fao.& (i111.81)

(cut off by bound) (cut off by bound)

Fig. 2. Implicit enumeration tree.

“allocation” variables as shown in Table 6 (rounded to two decimal places). The corresponding tabu search solution has the objective function value 1111 .Ol with the following solution (hub: allocation)

4: 4,11,15; 8: 8,12,19; 13: 7,10,13,16; 20: 1,2,3,5,6,9,14,17,18,20.

Using the information from LP and TS solutions (e.g.: node 11 is not a hub in the TS solution, in LP solution node 20 has many partial hub allocations) we generated an implicit enumeration tree as shown in Fig. 2.

The solution again confirmed the optimality of the tabu search heuristic solution.

4. Conclusions and future directions for research

In this paper we proposed new formulations for the multiple and single allocation p-hub median problems. The formulations have very tight linear programming relaxations: for the problems used in this study, LP solutions were integer in 96% in- stances. Even when the LP solution was not integral, it generated lower bound less than 1% (respectively, 0.1%) below the optimal value of the multiple (re- spectively, single) allocation p-hub median problem.

592 D. Skarin-Kapov et al./European Journal of Operational Research 94 (1996) 582-593

Table 6 Nonzero values of allocation variables in the p-HM-SA model with n = 20, p = 4, a = 1.0

2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 0.33 0.67 2 0.33 0.67 3 0.33 0.67 4 1 5 0.33 0.67 6 0.33 0.67 7 0.33 0.67 8 0.33 0.67 9 0.33 0.67 10 0.33 0.67 11 0.33 0.67 12 0.33 0.67 13 0.33 0.67 14 0.33 0.67 15 1 16 0.33 0.33 0.33 17 0.33 0.67 18 0.33 0.67 19 0.33 0.67 20 0.33 0.67

For the more difficult single allocation p-hub me- dian problem, in addition to information generated via LP relaxation, we propose the use of existing heuristic solution as a guidance in adding integrality constraints when needed. This approach opens inter- esting possibilities for interplay between heuristic and optimal solutions for hub location problems.

The formulations discussed in this study are large, and after relaxing the integrality constraints, we still have to solve large and difficult LP problems. There- fore, to solve optimally problems larger than the ones solved in this study will require larger computer resources. Another alternative is to use a heuristic method. Based on the results of this study, the tabu search algorithm of Skorin-Kapov and Skorin-Kapov (1994) could be used with a reasonable confidence on larger problems. Namely, for all the problems in this study, their heuristic achieved optimal solutions.

Based on the tight LP relaxations, one can use the sensitivity analysis theory of LP to obtain an insight to the sensitivity of a given solution with respect to, say, changes in the objective function coefficients

(e.g., changes in the work).

flows among nodes in a net-

Acknowledgements

The authors would like to acknowledge that re- search reported here was partially supported by NSF grant DMS-9218206. The research of Jadranka Sko- r-in-Kapov was also partially supported by NSF grant DDM-9307417.

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