Theory and Methodology

Bandwidth packing with queuing delay costs: Bounding andheuristic solution procedures

Ali Amiri a,1, Erik Rolland b,*,2, Reza Barkhi c,3

a College of Business, Weber State University, Ogden, UT 84408-3804, USAb Department of Accounting and Management Information Systems, Fisher College of Business, The Ohio State University,

Columbus, OH 43210, USAc Department of Accounting, Pamplin College of Business, Virginia Polytechnic and State University, Blacksburg, VA 24061-0101, USA

Received 1 November 1996; accepted 1 September 1997

Abstract

In this paper we propose a new formulation for the bandwidth packing problem (BWP) in telecommunications

networks. This problem is one of selecting calls, from a list of requests, to be routed in the telecommunications network.

We consider both revenue losses and costs associated with communications delay as parts of the objective. An e�cient

Lagrangean relaxation based heuristic procedure for ®nding bounds and problem solutions is demonstrated. Com-

putational results from a large array of instances are reported. We demonstrate that the procedure is e�cient in ®nding

good solutions while expending a modest amount of computational e�ort. Ó 1999 Elsevier Science B.V. All rights

reserved.

Keywords: Bandwidth packing; Path assignment; Call routing; Telecommunications networks; Lagrangean relaxation;

Sub-gradient search; Heuristics

1. Introduction

One important problem in managing a tele-communication network involves deciding whichcalls on a list of requests, called a call table,should be routed on the network, and subse-

quently determining a path for each call to berouted. The path for each routed call is to beselected from all possible paths in the network.Typically, the network topology, the capacities ofthe links, the call table (including revenue andtra�c requirements/demand of each call), and aunit delay cost are given. Versions of this problemhave been studied by Anderson et al. (1993),Laguna and Glover (1993), Cox et al. (1991),Parker and Ryan (1995) and Park et al. (1996),and this problem is typically referred to as thebandwidth packing problem (BWP). The objec-tive of the BWP has in these past research e�orts

European Journal of Operational Research 112 (1999) 635±645

* Corresponding author.1 E-mail: aamiri@weber.edu.2 E-mail: rolland.1@osu.edu.3 E-mail: reza@vt.edu.

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 4 0 1 - 3

been to maximize the total revenues from callsthat are routed.

Route (or path) selection is a signi®cant factorin determining response time experienced by net-work users, and has a major e�ect on the utiliza-tion of network resources (e.g., node bu�ers andlink capacities). A good routing policy could alsoallow new users to utilize the network withoutsigni®cant deterioration of the quality of service toexisting users and without incurring the costs ofestablishing new links or upgrading the capacitiesof existing links.

During the management process of the net-work, tradeo�s have to be made between revenuemaximization and response time to users. If therevenue maximization factor alone is considered,network users will experience signi®cant delays,and the quality of service will deteriorate. Themodel developed in this paper addresses thisproblem by incorporating both revenue loss andresponse time (total delay) costs in the objectivefunction.

Anderson et al. (1993), Laguna and Glover(1993), Cox et al. (1991), Parker and Ryan (1995)and Park et al. (1996), all proposed a version of theBWP that considers only revenue maximization.They formulated the problem and presented heu-ristic solution procedures based on tabu search(Anderson et al., 1993; Laguna and Glover, 1993),genetic algorithms (Cox et al., 1991), columngeneration (Parker and Ryan, 1995), as well asinteger programming (Park et al., 1996). Moti-vated by the important applications for bandwidthpacking, as well as by the limitations of the currentmethods, we present a new formulation that seeksto maximize the di�erence between total revenueof calls to be routed and total delay cost. We de-velop a procedure which generates feasible solu-tions as well as bounds for this problem.

The remainder of this paper is organized asfollows. In Section 2, a mathematical formulationof the path assignment problem is presented. ALagrangean relaxation of the problem, obtainedby dualizing a subset of the constraints, is pre-sented in Section 3. A heuristic solution procedureis developed in Section 4. Computational resultsare reported in Section 5. The conclusions aresummarized in Section 6.

2. Problem formulation

In order to formulate the BWP in a telecom-munications network, we introduce the followingnotation:

The BWP can now be de®ned as follows:Given a graph G� (N, E) and a set of call re-

quests (a call table) M� (O, D, r, d), we seek tomaximize the pro®ts from the routed calls, whileminimizing the queuing delay costs, and while notexceeding the capacities on the communicationlinks.

We assume that the network topology, thecapacity of the links, and the tra�c requirementsand revenues for all the calls are known. We alsomake some assumptions which are typically usedin modeling the queueing phenomena in tele-communications networks. Speci®cally, we as-sume that nodes have in®nite bu�ers to storemessages waiting for transmission on the links,that the arrival process of messages to the net-work follows a Poisson distribution, and thatmessage lengths follow an exponential distribu-tion. We further assume that the propagationdelay in the links is negligible, and that there isonly a single class of service for each communi-cating node pair.

Even though the list of calls (requests) is as-sumed to be known in advance, the tra�c re-quirement for each call is usually bursty, as is thecase with video and data transmission. That is,calls exhibit variable bit rates. As an approxima-tion, we will use the M/M/1 model for linkqueueing delays. The validity of this assumption issupported by experimental evidence (in the designof packet-switched networks): the optimal routingis insensitive to the shape of the delay versus link

N the set of nodes in the networkE the set of undirected links in the networkM the set of calls. Each call is represented by a

communicating node pairdm the demand of call m 2 Mrm the revenue from call m 2 MO(m) the source node for call m 2 MD(m) the destination node for call m 2 MQij capacity of link (i,j)C unit delay cost

636 A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

load curve, and is only a�ected by its asymptoticvalue, i.e. the link capacity (which is generally thesame for all models) (Gerla, 1973).

Under these assumptions, the telecommunica-tions network is modeled as a network of inde-pendent M/M/1 queues (Kleinrock, 1964, 1976), inwhich links are treated as servers with service ratesproportional to the link capacities. The customersare messages whose waiting areas are the networknodes. The queueing delay in link (i, j) is 1/(lQij ÿ Aij), where 1/l is the average messagelength, Qij is the capacity of link (i, j), and Aij isthe arrival rate of messages to link (i, j). The av-erage end-to-end delay in the network can be es-timated as the weighted sum of the expected delaysof the links in the network.

We de®ne the decision variables as follows:

Y m � 1 if call m is routed;

0 otherwise;

�

X mij �

1 if call m is routed through a path

that uses link �i; j�;0 otherwise;

8><>:

W mij �

1 if call m is routed through a path

that uses link �i; j� in the

direction of i to j;

0 otherwise:

8>>><>>>:Now, the average end-to-end delay becomes:

1

T

X�i;j�2E

Pm2M xm

ij

Qij ÿP

m2M xmij;

where

T � 1

m

Xm2M

Am:

Note that T is constant, and therefore can be re-moved from consideration. Now, since C is de®nedas the unit queueing delay cost, the total queueingdelay cost is represented by the term:

CX�i;j�2E

Pm2M dmX m

ij

Qij ÿP

m2M dmX mij:

The organization managing the network incursan opportunity cost associated with this queuingdelay. This opportunity cost could represent thelost revenues from customers who ®nd the delayunacceptable, and hence switch to competitors.Alternatively, seen from a customer viewpoint, thedelay cost could be viewed as the cost of lostproductivity or the cost of delayed decision-mak-ing. Thus, the delay costs can be viewed as beingsimilar to the backorder costs in a manufacturing/inventory system; that is, the delay cost representsan expression of the potential loss of customers,customer loyalty or customer productivity. In thispaper, the objective is to maximize the revenues ofrouted calls less this backorder, or queueing delay,cost. Given our objective of minimizing the delaycosts while maximizing revenues, we can introducearti®cial delay costs to achieve a desired level oftradeo� between total revenues and average re-sponse time to customers. Our proposed modelexplicitly addresses this tradeo�, and some impli-cations of this tradeo� are further discussed inSection 5.

The problem can now be properly formulatedas follows:

Problem P:

ZP � maxXm2M

rmY m ÿ CX�i;j�2E

Pm2M dmX m

ij

Qij ÿP

m2M dmX mij

�1�s.t.Xj2N

W mij ÿ

Xj2N

W mji �

Y m if i � O�m�

ÿY m if i � D�m�

8�i; j� 2 E and m 2 M ;

0 otherwise

8>>>>>>><>>>>>>>:�2�

W mij � W m

ji 6X mij ; 8�i; j� 2 E and m 2 M ; �3�X

m2M

dmX mij 6Qij; 8�i; j� 2 E; �4�

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645 637

X mij 2 f0; 1g; 8�i; j� 2 E and m 2 M ; �5�

Y m 2 f0; 1g; 8m 2 M ; �6�

W mij 2 f0; 1g; 8�i; j� 2 E and m 2 M : �7�

In this formulation, the two terms in the objectivefunction represent total revenues of routed callsand the total queuing delay cost, respectively. Theformat of the queuing delay component is in ac-cordance with our queuing modeling assumptions.The queuing delay cost (C) implicitly measures theimportance of network utilization (higher C meanslower network utilization). We address the issue ofthe choice of C further in our computational re-sults. As the queuing delay costs are to be mini-mized, we assume that the denominator is alwaysgreater than zero. Constraint set (2) contains the¯ow conservation equations which de®ne a route(path) for each call represented by a communi-cating node pair. Constraint set (3) links togetherthe X m

ij and W mij variables. Even though the prob-

lem can be correctly formulated with either X mij or

W mij variables only, both of them are useful in the

Lagrangean relaxation developed in the next sec-tion. Constraint set (4) represents the capacityconstraints on the links. Constraint sets (5)±(7)enforce the integrality conditions on X m

ij ,Ym andW m

ij variables, respectively.

3. A Lagrangean relaxation of the problem

Problem P is a combinatorial optimizationproblem with a nonlinear objective function. Theproblem studied in Anderson et al. (1993), Lagunaand Glover (1993) and Park et al. (1996) is a spe-cial case of problem P and is known to be NP-complete (Garey and Johnson, 1979). Since prob-lem P is a nonlinear mixed integer programmingproblem and since a typical problem containsthousands of variables and constraints, it is di�-cult to solve this problem to optimality usingstandard mixed-integer±linear programming tools.We propose, instead, a composite upper and lowerbounding procedure based on a Lagrangean re-laxation of the problem. Consider the Lagrangeanrelaxation of problem P obtained by dualizing

constraint set (3) using non-negative multipliers amij

for all �i; j� 2 E and m 2 M , respectively.

Problem L:

ZL � maxXm2M

rmY m ÿ CX�i;j�2E

Pm2M dmX m

ij

Qij ÿP

m2M dmX mij

�Xm2M

X�i;j�2E

amij�X m

ij ÿ W mij ÿ W m

ji � �8�

s.t. Eqs. (2), (4)±(7).Problem L can be decomposed into two sub-

problems as follows:

Problem L1:

maxXm2M

rmY m ÿXm2M

X�i;j�2E

amij�W m

ij � W mji � �9�

s.t. Eqs. (2), (6) and (7).

Problem L2:

maxXm2M

X�i;j�2E

amijX m

ij ÿ CX�i;j�2E

Pm2M dmX m

ij

Qij ÿP

m2M dmX mij

�10�s.t. Eqs. (4) and (5).

Problem L1 can be further decomposed into |M|sub-problems (one for each call) as follows.

max rmY m ÿX�i;j�2E

amij�W m

ij � W mji � �11�

s.t.

Xj2N

W mij ÿ

Xj2N

W mji �

Y m if i � O�m�ÿY m if i � D�m�

8i 2 N ;

0 otherwise

8>>><>>>: �12�

Y m 2 f0; 1g; �13�

W mij 2 f0; 1g; 8�i; j� 2 E: �14�

Similarly, problem L2 can be further decomposedinto |E| sub-problems (one for each link) as fol-lows.

638 A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

maxXm2M

amijX

mij ÿ C

X�i;j�2E

Pm2M dmX m

ij

Qij ÿP

m2M dmX mij�15�

s:t:Xm2M

dmX mij 6Qij; �16�

X mij 2 f0; 1g 8 m 2 M : �17�

We note that the Lagrangean problem (L) doesnot satisfy the integrality property, because thelinear programming relaxation of (L) does notnecessarily have an integer solution. Thus, ourrelaxation of problem (P) can theoretically give anupper bound which is at least as good as, andpossibly better than the linear programming re-laxation of (P).

Each sub-problem of L1 for a call m can besolved optimally by ®nding the shortest path (e.g.,Ford and Fulkerson, 1962) problem from O(m) toD(m) using the non-negative multipliers am

ij as thelink costs. If the revenue from the call is greaterthan the cost of that shortest path (i.e., rm > am

ij�,then the call is routed through that path. If not,the call is not routed, and we set Ym� 0 andW m

ij � 0; 8�i; j� 2 E:Each sub-problem of problem L2 correspond-

ing to a link (i, j) is equivalent to a single con-straint (0,1) knapsack problem with a nonlinearobjective function. We have chosen, however, torelax the integrality constraints and solve thecontinuous version using the following greedyprocedure:

Procedure Proc1:Step 1: Reorder the X m

ij variables by sortingthem in non-increasing order of am

ij=dm; re-indexthe variables in this order, and let m� 0.

Step 2: Let m�m + 1 and set

X mij �

X0 if amij > 0 and X0 > 0;

0 otherwise

�where

X0 � min 1;1

dm Qij ÿ Sÿ �248<: ÿ CdmQij

amij

!1=2359=;

and S �Xk<m

dkX kij:

Step 3: If m� |M| stop; if X mij < 1 then stop and

set X kij � 0 for k � m� 1; . . . ; jM j. Otherwise go to

step 2.

4. A heuristic solution procedure

Using the Lagrangean relaxation presented inSection 3, we can generate feasible solutions aswell as lower bounds for the optimal solution ofproblem P. The success of this approach dependsheavily on the ability to generate good Lagrangeanmultipliers.

If we let ZL(a) be the value of the Lagrangeanfunction with a multiplier vector (a), then the bestbound using this relaxation is derived by calcu-lating ZL�a�� �Min�a�fZL�a�g. The challenging is-sue in generating good bounds using Lagrangeanrelaxation is to ®nd a good set of multipliers. Thisis in general known to be a very di�cult task(Gavish, 1978). In practice, a good but not nec-essarily optimal set of multipliers is often derivedusing iterative methods such as sub-gradient op-timization method and various multiplier adjust-ment methods known as ascent (descent) methods(Bazaraa and Goode, 1979). In this study we usethe sub-gradient optimization method to searchfor ``good'' multipliers. The sub-gradient methodis a modi®ed version of the gradient method inwhich sub-gradients replace gradients (Gavish,1978). Since this method is well understood, we donot provide its implementation details; we sum-marize however the particulars of our algorithm inAppendix A. Further theoretical and computa-tional properties of the subgradient method arediscussed in Fisher (1981), and in Held et al. (1974)

In this section we outline a heuristic procedureto solve problem P. Procedure-Feas attempts togenerate a feasible solution to problem P at everyiteration of the sub-gradient optimization algo-rithm using information provided by the solutionto problem L1. The best feasible solution is re-tained when the sub-gradient algorithm is termi-nated. Note that in the solution to problem L1every call is either routed through the links in thenetwork or not routed at all. However, there maybe some links with loads higher than their avail-able capacities. This simple heuristic attempts to

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645 639

route calls through the network without exceedinglink capacities. Thus, the heuristic guarantees togenerate a feasible solution at every iteration of thesub-gradient optimization procedure. The com-plete heuristic is stated below.

Procedure FEAS:Step 1: Order the calls by sorting them in non-

increasing order of the optimal values of the ob-jective functions of their corresponding sub-prob-lems of problem L1. Start with the ®rst call.

Step 2: Check if the call can be routed using thepath determined in the solution of its corre-sponding sub-problem of problem L1 without ex-ceeding the available link capacities. If yes, the callis routed only if its revenue exceeds the increase inthe queueing delay cost as a result of it beingrouted through the network. Update the availablelink capacities if the call is routed. Proceed withthe next call, and repeat step 2. Stop when all callshave been examined.

5. Computational results

The solution procedures presented in Section 4were coded in Pascal. A number of computationalexperiments were performed using an IBM-3090model 2000 running under MVS/XA version 2.2.3.To evaluate the e�ectiveness of those procedures,we used nine sets of networks with 10±50 nodes (insteps of 5) generated randomly, but systematically,to capture a wide range of problem structures. Foreach network set (10; 15; . . . ; 50 nodes), we gener-ated ®ve problem groups, where only the numberof calls di�er (see Table 2). The delay coe�cientwas kept constant at 1 unit. In order to achieve areasonable level of con®dence about the perfor-mance of the solution procedure versus the prob-lem structure, we generated ten instances byrandomly creating di�erent call tables for eachinstance (that is, each line in Table 2 reports theaverage results from ten problem instances).

To study the e�ect of the queuing delay cost bykeeping the number of calls constant, we generateda new set of problems (Table 3) by varying thequeueing delay costs from 0.5 to 20 (the problemgroups are based on one problem instance from

Table 2, where P� 60 (see below for a discussionof P). For each problem, we randomized the calltables to create ten problem instances (each line inTable 3 reports averages from these ten probleminstances). We solved all the 450 problems in bothproblem sets, and thus a total of 900 problem in-stances were solved.

The following test problem generator was usedto generate the networks above: First, the gener-ator locates the speci®ed number of nodes on a100 ´ 100 grid. Each node has a degree equal to 2,3 or 4 with probability of 0.6, 0.3 and 0.1, re-spectively. We repeat the following procedure foreach node i 2 N : Determine node i's closestneighbor (in terms of Euclidean distance) withunsatis®ed degree requirement, label this node j.Add arc (i, j) and repeat this until ®rst, node i'sdegree requirement is satis®ed or all the nodes withunsatis®ed degree requirements have been consid-ered. In the latter case, connect node i to its closestneighbors to which it is not already connecteduntil the degree requirement of node i is satis®ed.At the end check if the network is connected; ifnot, add links necessary to make it connected.Each link in the network is randomly assigned acapacity equal to 48, 96, 192, or 500 with equalprobabilities. The average number of links foreach of the networks are shown in Table 1.

The test problem generator also produced thecall tables. The call table contains informationabout the origin and destination nodes for allcommunicating node pairs, as well as the revenuesand communication demand for those calls. Thenumber of calls (P) in the tables varies between

Table 1

Average number of links in the networks

Number of nodes in the

network

Av. number of links in the

networks

10 14.6

15 21.6

20 28.6

25 36.8

30 43.8

35 52.2

40 59.8

45 68.4

50 75.2

640 A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

Table 2

Computational results with di�erent percentages for the number of available calls

|N| P C Average

Total Delay % link utillization CPU

% gap revenue cost Average Maximum

10 50 1 4.18 511.6 26.6 45.2 85.8 10

10 60 1 2.84 610.8 37.2 52.8 91.4 12

10 70 1 3.08 654.4 31.8 54.0 89.8 16

10 80 1 3.86 723.2 42.8 61.8 89.4 18

10 90 1 4.46 757.2 36.6 60.2 88.6 21

15 50 1 2.86 1007.8 68.2 63.8 93.0 42

15 60 1 3.10 1148.8 77.4 68.4 93.4 50

15 70 1 3.32 1253.4 88.8 71.0 91.6 59

15 80 1 3.12 1351.4 96.0 72.8 93.2 43

15 90 1 3.68 1428.8 98.2 72.8 91.8 88

20 50 1 3.34 1465.8 113.8 65.2 94.8 112

20 60 1 3.34 1598.6 126.0 66.6 95.4 134

20 70 1 3.28 1706.6 129.0 67.6 95.2 154

20 80 1 3.42 1754.6 131.6 67.8 94.6 185

20 90 1 3.50 1843.6 139.8 71.2 95.8 206

25 50 1 4.76 1727.0 122.0 61.2 94.8 255

25 60 1 4.80 1897.2 149.0 63.8 95.2 295

25 70 1 5.22 2099.8 182.6 68.2 96.0 360

25 80 1 4.92 2275.2 169.6 68.2 95.2 405

25 90 1 4.54 2419.0 179.4 69.6 95.6 465

30 50 1 4.20 2198.0 162.8 64.0 93.8 467

30 60 1 4.32 2438.0 194.8 65.8 95.6 576

30 70 1 4.56 2603.2 189.2 66.2 95.8 660

30 80 1 4.32 2824.2 211.6 69.2 95.8 762

30 90 1 4.56 2969.4 497.0 68.2 95.8 901

35 50 1 5.88 2525.2 197.0 62.4 95.2 799

35 60 1 5.02 2801.6 212.0 64.8 95.4 903

35 70 1 4.76 3099.2 220.2 66.4 94.8 1145

35 80 1 4.54 3342.4 248.8 69.2 95.6 1317

35 90 1 4.66 3509.6 255.8 69.4 95.6 1521

40 50 1 6.10 2841.4 214.0 63.2 94.4 1345

40 60 1 5.78 3097.6 213.2 65.2 95.0 1671

40 70 1 5.58 3358.0 279.6 69.0 96.0 1842

40 80 1 5.44 3570.20 293.4 70.0 96.0 2118

40 90 1 5.22 3821.6 296.0 71.8 95.4 2498

45 50 1 6.04 3491.0 314.2 65.8 96.6 2160

45 60 1 5.68 3765.0 314.8 67.0 95.2 2671

45 70 1 5.74 4008.6 322.4 68.6 96.4 2984

45 80 1 5.66 4249.2 328.8 69.4 96.0 3376

45 90 1 5.80 4387.5 311.8 68.5 96.3 3593

50 50 1 5.16 3844.4 312.0 65.2 96.0 3240

50 60 1 5.48 4097.8 337.3 65.8 96.3 3871

50 70 1 5.60 4353.0 346.8 66.5 96.5 4235

50 80 1 5.77 4454.7 366.3 66.3 96.3 4749

50 90 1 5.33 4664.3 351.7 67.3 96.3 5124

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645 641

Table 3

Computational results with di�erent unit delay costs

|N| P C Average

% Total Delay % link utilization CPU

gap revenue cost Average Maximum

10 60 0.5 3.32 615.4 21.4 54.0 92.6 12

10 60 1 2.84 610.8 37.2 52.8 91.4 12

10 60 5 2.48 672.4 92.2 44.4 78.2 12

10 60 10 3.04 529.4 121.2 36.8 67.0 14

10 60 15 4.00 502.0 147.2 32.2 64.4 12

10 60 20 6.46 463.2 154.8 27.2 63.0 11

15 60 0.5 3.24 1162.0 45.8 70.4 93.6 48

15 60 1 3.10 1148.8 77.4 68.4 93.4 50

15 60 5 2.20 1058.6 190.6 57.8 81.4 54

15 60 10 2.52 970.2 252.8 49.8 73.4 51

15 60 15 3.08 879.0 268.2 40.8 65.2 47

15 60 20 3.60 808.2 277.2 34.6 58.8 52

20 60 0.5 3.28 1627.6 84.6 69.2 96.4 138

20 60 1 3.34 1598.6 126.0 66.6 95.4 134

20 60 5 3.14 1411.0 219.6 52.6 83.6 129

20 60 10 2.46 1337.6 333.6 47.2 76.8 141

20 60 15 2.90 1247.0 388.2 42.8 67.8 135

20 60 20 3.64 1154.8 410.0 37.2 63.6 137

25 60 0.5 5.40 1916.4 93.6 67.0 96.4 284

25 60 1 4.80 1897.2 149.0 63.8 95.2 295

25 60 5 2.76 1720.8 300.0 53.8 83.2 297

25 60 10 2.96 1569.5 391.4 46.0 75.8 291

25 60 15 3.12 1438.6 427.2 38.8 68.8 287

25 60 20 3.36 1358.6 476.4 34.6 63.4 301

30 60 0.5 4.28 2500.6 142.2 69.4 97.0 587

30 60 1 4.32 2438.0 194.8 65.8 95.6 576

30 60 5 2.70 2188.2 363.8 54.2 84.0 554

30 60 10 1.86 2040.2 512.0 48.0 76.0 571

30 60 15 2.00 1897.0 588.2 41.8 70.6 598

30 60 20 2.80 1770.8 636.6 37.4 64.0 568

35 60 0.5 5.54 2858.2 152.6 67.6 96.8 881

35 60 1 5.02 2801.6 212.0 64.8 95.4 903

35 60 5 2.98 2519.6 414.0 53.0 85.2 914

35 60 10 1.98 2352.6 586.6 47.2 74.8 908

35 60 15 2.30 2191.2 680.6 41.0 69.0 894

35 60 20 2.96 2023.2 715.4 35.6 64.2 907

40 60 0.5 6.48 3161.6 161.2 68.2 97.8 1694

40 60 1 5.78 3097.6 231.2 65.2 95.0 1671

40 60 5 3.36 2818.4 500.4 54.8 86.2 1654

40 60 10 3.08 2610.0 701.8 48.2 79.2 1701

40 60 15 3.36 2390.6 780.2 41.2 70.8 1649

40 60 20 3.90 2204.6 823.4 36.0 65.4 1667

45 60 0.5 6.14 3844.4 220.6 70.2 97.6 2712

45 60 1 5.68 3765.0 314.8 67.0 95.2 2671

45 60 5 3.24 3386.4 609.2 56.0 84.8 2674

45 60 10 2.56 3113.8 820.8 48.4 78.0 2657

642 A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

50% and 90% of all possible calls. For example, ina 50 node network, using a percentage of 60(P� 60), there are 1470 calls (or 50 ´ 49 ´ 0.6). 4

For each call, the generator randomly determinesan origin and a destination node. The tra�c re-quirement and revenue for each call are generatedrandomly from two uniform distributions between20 and 40 and between 10 and 50, respectively. Theunit delay cost varies between 0.5 and 20 and it isset to 1 for the cases in Table 1.

Tables 2 and 3 show the average performancemeasures for di�erent networks over nine problemtypes. The results of the experiments are describedby providing the number of nodes in the network(|N|), the percentage of the number of calls (P), theunit delay cost (C), the average gap between the``best'' feasible solution value and the upper boundexpressed as a percentage of the upper bound, theaverage total revenue, the average total queuingdelay cost, as well as the average and maximumlink utilization.

Table 2 shows the results for di�erent percent-ages of the total number of available calls. Whenthis number increases, total revenue of routed callsincreases as a result of higher utilization of thenetwork links to serve a larger number of calls.However, when the number of available calls in-

creases, the average link utilization and totalqueueing delay cost increase (even though the unitdelay cost remains constant), indicating a deteri-oration in response time to users. This can be ex-plained by the dominance of the revenuecomponent in the objective function of the prob-lem over the queueing delay cost component. Theaverage gap between the feasible solution valueand upper bound varies between 2.84% and 6.10%with a mean of 4.60%. The CPU time (in seconds)varies between 10, for networks with 10 nodes, and5124 for networks with 50 nodes, with an averageCPU time of 1277 s.

The e�ects of changes in unit delay cost arereported in Table 3. Total revenue of routed callsdecreases when the unit delay cost increases asfewer calls are selected to be routed, to providebetter response time to users. The improvement inresponse time is re¯ected in the decrease in theratio of queueing delay cost by unit delay cost. Forexample, for the 50 node networks, this ratio onaverage decreases from 457.6 when the unit delaycost is 0.5 to 50.9 when this unit delay cost is 20.The improvement in response time is also indi-cated by the decrease in average and maximumlink utilization. For example, for the 50 nodenetworks, the average and maximum link utiliza-tion drop from 68% to 35.5% and 97.7% to 68.5%,respectively. The average gaps between feasiblesolution values and upper bounds vary between1.86% and 6.48% with a mean of 3.56%. The CPUtime (in seconds) varies from 11, for networks with10 nodes, to 3903 for networks with 50 nodes, withan average of 1133 s.

4 For reasons of comparison, we mention that Anderson et

al. (1993) reported results of computational experiments con-

ducted using networks ranging in size from 14 nodes and 35

calls to networks with 192 nodes and 20 calls. Park et al. (1996)

report results for problems with up to 30 nodes and 75 arcs, and

with a maximum of 75 calls.

Table 3 (Continued)

|N| P C Average

% Total Delay % link utilization CPU

gap revenue cost Average Maximum

45 60 15 3.12 2857.4 921.2 42.2 71.2 2683

45 60 20 3.52 2616.4 945.8 35.6 66.8 2689

50 60 0.5 5.68 4166.3 228.8 68.0 97.8 3841

50 60 1 5.48 4097.8 337.3 65.8 96.3 3872

50 60 5 3.05 3653.0 597.8 53.0 85.3 3903

50 60 10 2.30 3430.0 885.3 47.8 79.0 3877

50 60 15 2.53 3164.3 991.3 42.0 72.8 3861

50 60 20 3.23 2900.5 1019.3 35.5 68.5 3887

A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645 643

We conducted additional computational ex-periments to test the performance of the solutionprocedure when C is set to zero (that is, we ignorethe queueing delay). The average results are pre-sented in Table 4. From this table we see that ourprocedure is fairly e�ective even for the problemwhere no delay cost is considered. The averageduality gap in the case with no delay cost is 7.90%.

The convergence rate of the Lagrangean relax-ation is relatively stable for all test problems. Thechart below (Fig. 1) was generated using the tenproblem instances with number of nodes equal to30, percentage number of calls (P) equal to 60, andunit delay cost equal to 1. It shows the averagefeasible solution and upper bound values as afunction of the number of iterations. As shown bythe ®gure, the convergence rate is very high in the®rst 200 iterations (the gap is only 5.8% at the end

of the 200th iteration) but becomes relatively slowafter that. Fig. 1 is typical of all test problems.

6. Conclusions

In this paper we have provided a formulation,as well as a bounding and e�cient solution pro-cedure for the BWP. Our research contributionsare as follows: (i) we proposed a new formulationfor the BWP; (ii) we considered both revenue lossand costs associated with delay as an integral partof the objective function; and ®nally (iii) we pro-posed both a bounding and solution procedure forthis problem. Our solution procedure considers allpossible paths for routing the calls (as opposed tojust a subset of the possible paths). In our com-putational experiments we demonstrated that, onthe average, our heuristic produced consistentlygood solutions, with average bounds of 4.6% forthe unit delay cost case and 3.56% in the case ofvarying delay costs. Further, the heuristic requiredonly a modest computational e�ort, with averageCPU times of about 1200 s for the 900 di�erentproblem instances solved.

Appendix A

Given an initial multiplier vector a0 (set to thezero vector in this study), a sequence of multipliersis generated by updating the vector at the iterationk using the formula

ak�1 � ak � tk�W k ÿ X k�;

Table 4

Average computational results with no unit delay costs

n P C % Gap Total revenue Delay cost Average link

utilization

Maximum link

utilization

CPU

10 60 0 8.26 652 0.0 65.2 93.8 12

15 60 0 7.48 1015 0.0 62.4 98.0 47

20 60 0 7.92 1658 0.0 70.2 98.6 151

25 60 0 8.14 2070 0.0 65.0 98.8 294

30 60 0 7.68 2607 0.0 68.0 99.0 591

35 60 0 8.06 3097 0.0 68.4 98.8 877

40 60 0 7.74 3352 0.0 67.6 99.0 1673

45 60 0 8.28 3845 0.0 69.8 98.8 2635

50 60 0 7.58 4190 0.0 68.6 99.2 3901

Average 7.9

Fig. 1. Convergence of the Lagrangean relaxation.

644 A. Amiri et al. / European Journal of Operational Research 112 (1999) 635±645

where ak�1 and ak are the multiplier vectors at it-erations k + 1 and k respectively, (Wk,Xk) is partof the optimal solution to the Lagrangean ProblemL with multiplier vector ak and tk is a positivescalar step size.

It is well known that lim sup ZL(ak) convergesto ZL(a*) if tk ! 0 and

P1k�0tk !1 (Poljack,

1967). Since in general these conditions are verydi�cult to satisfy, the sub-gradient optimizationmethod is always used as a heuristic. In this study,we used the following step size that has been foundto be satisfactory in practice:

tk � kk�ZL ÿ Zf �ak��=jjW k ÿ X kjj2;where Zf is the value of the best feasible solutionfound so far and kk is a scalar satisfying 06 kk 6 2.This scalar is set to 2 at the beginning of the al-gorithm and is halved whenever the bound doesnot improve in 20 consecutive iterations. The al-gorithm is terminated after a speci®ed number ofiterations (set equal to 500 in this study) unless anoptimal solution is reached before that point. Thealgorithm is also terminated if the gap between thebest upper bound and the best feasible solutionfound is less than 0.01% of the best upper bound,or the best upper bound does not improve in 100consecutive iterations by at least 0.01%.

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