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A heuristic approach for capacity expansion of packet networks
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 91 (1996) 395-410
Theory and Methodology
A heuristic approach for capacity expansion of packet networks Amitava
Dutta
a,., Young
Ki Kim b,..
a Department of Decision Sciences and MIS, College of Business, George Mason University, 4400 University Drive, Fairfax, VA 22030-4444, USA b Department of Management Science, College of Business, University oflowa, Iowa City, IA 52242, USA
Received March 1994; revised February 1995
Abstract
We address the problem of expanding transmission capacity of an existing packet network over a multiperiod planning horizon, the objective being low total cost of expansion. Discrete capacity choices, interaction with routing decisions, and economy of scale in the cost of capacity make it extremely difficult to decide when, where and how much capacity to add. A fast heuristic solution method is developed based on the well established Flow Deviation routing algorithm. The heuristic begins by making myopic expansion decisions, which are then subsequently adjusted to account for economies of scale in the cost of capacity. Heuristic solutions are compared to a benchmark which approximates the real cost function by its linear lower envelope. Since the number of possible expansion plans is an exponential function of the number of edges, capacity choices, and periods in the planning horizon, a fast heuristic allows one to look beyond small problems at more realistically sized ones. Keywords: Heuristics; Communication; Capacity expansion
1. Introduction
The d e m a n d for telecommunications services has exploded in recent times. Technological advances have m a d e it cheaper to communicate electronically. The decreasing cost of computing power has also led to the distribution of computing activities and a need to communicate among different computing agents. While voice has been and still continues to be the dominant type of traffic, the importance of data applications is on the rise. One prediction estimates that data and * Corresponding author. Author's current address: SongPa-Gu, GaRak-Dong, 1915 Seoul, South Korea. **
voice will constitute 61% and 39% of private network traffic respectively, by 1995 [18]. Applications such as electronic mail, C A D / C A M , order entry, financial and accounting inquiries have contributed to this increase in data volume. We will not elaborate further on these trends as they are widely recognized. For some recent examples of trends see [1]. In this paper, we are interested in one important consequence, which is the need to expand network transmission capacity in response to anticipated increase in demand. Since capacity is expensive, cost minimization is an important objective in generating expansion plans. Consider a hypothetical wide area network shown in Fig. 1. The nodes represent origins/destinations for traffic. Over time, the traffic demands of
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00089-5
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A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
existing origin-destination (O-D) pairs may increase a n d / o r new nodes may be added to the network, requiring additional capacity. Fig. 2 shows the cost structure of transmission capacity, the two characteristics being (i) discreteness and (ii) economy of scale. The latter implies that it may be beneficial to install large expansion options to meet future demand, rather than several small options as needed [10]. Thus a multiperiod view of the expansion problem is needed. Fig. 1 also shows that capacity and routing decisions will be closely related, and there are many alternatives for the latter. These characteristics make the expansion problem an intractable one. A single expansion plan consists of a capacity assignment for each edge of the network for each period. Even for a small network of 10 nodes, 20 edges, 4 capacity choices and a 5 period planning horizon, there will be approximately 10 60 expansion plans. In fact the problem remains intractable for H = 1, since this single period problem is a more general case of the capacitated minimum spanning tree, which is known to be NP-hard [13].
1.1. Related literature A large number of heuristics exist for single period capacity assignment. They cope with complexity by iterating between capacity and routing decisions until some appropriate stopping criteria are met [2,4,7,8,11,12]. These heuristics are quite
+130 140 ~
200 Fig. 1. A widearea network.
~. 100
C 0 $
t
I Capacity Fig. 2. Generalcost structureof capacity.
fast and their effectiveness has been established through extensive testing with many and varied problem instances, rather than through comparison with lower bounds. The integer programming model in [6] is an exception in that it jointly optimizes routing and capacity and also produces a lower bound for comparison. Computation time is not reported in [6]. Literature on the multiperiod expansion problem addressed in this paper is somewhat more scarce. While heuristic approaches also predominate here, considerable simplifying assumptions have been made. For instance, the cost function was assumed to be a concave function of link capacity in [16]. In [17] traffic for communicating node pairs was routed over the same path for all time periods, ignoring the interplay between capacity and routing. The 'deferral' strategy proposed in [15] postpones installation of additional capacity for as long as possible by routing traffic around saturated links, thereby ignoring economy of scale benefits. The integer programming formulation in [3] does not make such simplifying assumptions. However, while solution quality is good, computation time increases sharply with problem size. In short, we have efficient heuristics for capacity expansion which make major simplifications of problem characteristics, and optimization models which are more realistic but can only be solved approximately, and with considerable computational effort. In this paper we develop and test a heuristic that can produce low cost expansion plans for large problem instances within reasonable computation times, without having to make gross
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
397
simplifications about major problem characteristics such as the cost structure of capacity or routing policies. While a lower bound cannot be conveniently computed, a benchmark problem is described to judge quality of the heuristic solutions.
Objective:
2. Problem statement and characteristics
2.1. Cost of capacity and timing of expansions
We now give a more precise statement of the expansion problem and assumptions about different components thereof. Major characteristics, such as discrete cost, have not been simplified to make the problem tractible. However, as with any abstraction of a physical problem, some approximations by omission and commission have been made in the problem statement. Their impact on the applicability of the heuristic solution method is discussed.
Note that we start with existing link capacities rather than assume links to have zero initial capacity. In practice, one often does not have the opportunity to plan capacity expansions from scratch. We assume that expansion decisions are made periodically rather than continuously. This is quite realistic in most circumstances. However, we also assume that the extra capacity for a period is installed instantaneously at the start of the period. Although this assumption is not quite realistic, and results in some cost overestimation, it is commonly made in the capacity expansion literature for reasons of tractability. The overestimation decreases as periods become shorter relative to the length of the planning horizon. We do not make any simplifying assumptions about discreteness of capacity or the presence of economies of scale. Typically, costs of capacity have fixed and variable components. The former is dependent on the type and size of link being installed, while the latter is usually tied to the physical distance covered. Therefore we retain the essential elements of cost structure in our problem.
Notation: : Number of periods in planning horizon. : Number of links in the network. : Traffic demand for O-D pair s in period t. Rst A max : Maximum performance limit on average delay. : Computed average network delay in peA, riod t. : Number of capacity expansion options. : Capacity of expansion option i, i = di 0,1 . . . . . ,7,(; d o = 0. P,j : Cost of installing expansion option i on link j : Total capacity on link i at start of period t Cit in the final plan. L, : Flow assigned to link i in period t. H
The capacity expansion problem can now be stated as follows: Given:
{ H,
Cio Vi, Rst Vs, t, di, i = 1 . . . . . ~e';
Pij, i = 1 . . . . . ,.,2r; j = 1 . . . . . c~}. Decide:
{Ci, and fit, i = 1,...,.o~; t = 1 . . . . . H } .
.~
Min
H
Y'~ ~ ( C i t - C
i,t_l)
i=1 t=l
subject to At
_~
2.2. Routing method Due to the relationship between capacity and routing decisions both are produced as outputs of our solution method. In particular, the route(s) assigned to the traffic for any specific O-D pair can change from period to period. However, many routing methods exist [14], and we have to make some choice among them. Dynamic routing methods respond to short term variations in network traffic by altering the paths used, while static methods do not. However, static routing is easy to describe and allows the direct computation of average delay for a given network capacity. Also,
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A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
it has been observed that at steady state, flow patterns and delays induced by good dynamic routing policies are very close to those obtained with optimal static routing [8]. It has therefore been common in the literature to employ static routing for capacity planning problems and we do so here as well. Two major variations of static routing are the bifurcated and nonbifurcated cases. Take node pair (1, 11) in Fig. 1 for example. If bifurcated routing were used, then in a given period, packets between 1 and 11 could be distributed over three paths say {1, 3, 12, 5, 11}, {1, 2, 4, 5, 11} and {1, 3, 4, 5, 11}. If nonbifurcated routing were employed, all packets between 1 and 11 would have to follow one path (in a period), say {1, 3, 4, 5, 11}. We use nonbifurcated routing as it tends to better utilize available network capacity. Other static routing algorithms could be easily substituted in its place within our heuristic.
2.3. Performance criterion The common performance criterion for a data network is average packet delay. Each link in the network is a server whose service rate is determined by its capacity, while packets routed on that link represent customers. This produces a network of queues and, under some assumptions that have been found to be quite robust, the average packet delay in period t, At, is commonly expressed as [9]
At = -3"i=1
fit "fit
(A t ~ oo if Cit
2.4. Limitations of problem statement The effect of assuming instantaneous capacity installation has already been discussed. In addition, the problem statement implicitly assumes that the number of nodes and topology of the network remain constant over time. This is not a serious limitation. New nodes introduced in future periods may be included from the start and assigned a traffic demand of zero for all periods in which they physically do not exist. Similar conditioning of data can be used to handle changes in topology where the number of nodes is unchanged but new links are added between existing nodes. Note also, that the problem statement implicitly assumes the nominal cost structure of capacity to remain fixed over time. This has no effect on the algorithm for our heuristic and has been made solely for implementation convenience to avoid having to store several arrays of costs, one for each period. If separate cost arrays are used for each period, it is possible to capture changes in nominal cost. However, these storage requirements can quickly become quite considerable. One assumption not readily apparent, is that capacity, once installed, is not removed for the remainder of the planning horizon. Physical removal is usually necessitated by aging or technical obsolescence and occurs far less frequently than the installation of additional capacity. Thus, for any given instance of the expansion problem, very few links would be candidates for removal. These links can be assigned a capacity of zero at the start of the appropriate period, and the heuristic can then assign a new capacity to them. In summary, the heuristic to be developed can, with some preconditioning of input parameters, be applied to several variations of the capacity expansion problem.
3. Solution method
Since the problem is NP-hard, we aim for good approximate solutions. Also, since we want reasonably quick approximations for large problems,
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
it is natural to see if efficient algorithms for single period capacity assignment, referenced in Section 1.1, can be adapted for the multiperiod problem.
Co
RI
Note that a good expansion plan cannot be obtained simply by applying existing single period heuristics once for each period in the planning
R,
R2
Period~
Period2
Periods
PeriOd4
Period5
Period6
6) Treat each perid in isolation. For each period i, use a single period heuristic to increase capacity from C Oto a vahie~ i which accommodates Ri while staying just below the allowed delay. Resulting sequence ~ . ."~6, is myopic since it does not consider economies of scale.
E1
~2
~3
C4
~5
C6
(ii)
From the sequence ~ - l . . . ~6. extract the six expansion values for any one link. Resehedule these six expansions considering economy of scale benefits. Repeat for each link in the network. Retain only the rescheduled assignments of all links for period 1, giving vector ~ .
col I
R2
R3
R5
R~
(iii)
In vector ~l, eliminate small expansion choices from selected links. This yields final capacity expansion decision for period 1, giving vector C 1.
colR i C1
R2
R3
C2 . .. C6 to be determined in like manner (iv)
Repeat steps (ii) through (iv) for i=2,...,6, using
399
Ci.I as the starting capacity.
Fig. 3. Heuristic for capacity expansion.
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A. Dutta, Y.K, Ka'm/ European Journal of Operational Research 91 (1996) 395-410
horizon. Such a procedure would ignore tradeoffs involving economies of scale in the cost of capacity. Further, single period heuristics generally assume zero initial capacities on links. Substantial modifications would be necessary to ensure that the capacity at the end of one period is the starting point for expansion in the next period. In short, adaptation from single to the multiperiod case involves several intricacies. 3.1 Overview of heuristic
In Fig. 3, we first present an informal overview of the major components of our heuristic. Algorithmic specification will appear subsequently. Fig. 3(i) shows some parameters of the expansion problem on a time line. Ri is the vector of traffic requirements of all O-D pairs in period i, while C O is the vector of capacities on different edges of the network at the end of period 0. We need to determine C1 . . . . . C6, the capacity vectors at the start of periods 1 through 6. The final capacity decision for each period i, Ci, is reached in three steps. First an initial expansion plan is generated where demand for each period is met independent of that in others. This is done using existing single period algorithms which are known to be fast. This myopic plan is then modified in the second step by rescheduling the expansion plan of each link independent of other links, in order to use economies of scale. This step can be viewed as being myopic in space since each link is treated in isolation. The third step attempts to utilize excess capacity installed on some links due to rescheduling, by eliminating small expansions on some links and rerouting their traffic through this spare capacity. This results in further cost reductions. Details of each step follow. 3.2. Step 1. Expansion - Each period in isolation
The basic operation consists of expanding Ci to accommodate the demand of some future period R j, j > i. To do so, Rj must first be routed 'judiciously' through C i to minimize average de-
lay. There can be three outcomes. The average delay may be less than the permissible limit, more than the limit, or it may be simply infeasible to route Rj through C i with finite delay. The last two outcomes will require expansion of capacities on an 'appropriately' selected collection of links. The Flow Deviation Algorithm (FDA) [5] is among the best established ones to judiciously route traffic demand through a given network capacity. Since details may be found in [5], we only summarize the workings of the FDA to motivate its use in our heuristic. Capacity and traffic requirements are given as inputs to the FDA. It starts from an initial feasible flow and attempts to iteratively adjust this flow in a steepest-descent manner, so as to reduce average delay, until no significant reduction in delay is possible. The rate of descent is measured by the ratio of change of delay with respect to change in flOW, (OA/~fi) , for each link i. The FDA is summarized as follows:
P R O C E D U R E FDA; Step 1. Find a feasible starting flow f 0 and set n~-0 Step 2. f n + l ~ (1 - h ) f n + hi7~ Step 3. IF ( A ( f ~) - A(fn+l)) < 0 STOP; ELSE n~n+l and go to Step 2; where A ( f ~) is average network delay under flow fin, v n is shortest route flow under the metric l k = (OA/Ofk) , A minimizes A((1 - A ) f n + AtTn), and 0 is an acceptable tolerance. END FDA;
The process of expanding C i to accommodate Rj begins by temporarily adding the largest and most expensive capacity option to each link in C i. FDA is used to route Rj through this network. (We are assuming that the largest available expansion option, when added to Ci, will be sufficient to meet Rj within the permissible delay limit. Otherwise the entire problem is infeasible.) The average delay for this overcapacitated network will probably be much lower than the permissible limit. So we iteratively choose smaller
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410 Table 3 Rescheduled plan - Adjustment for link 1
Table 1 Example expansion costs Expansion options
Capacity (Kbits/sec)
Cost (KS)
1
20 40 80
10 18 29
2 3
capacity expansions on selected links, until average delay increases to a point just under the permissible limit. Using a greedy approach, we would like each iteration to result in the least increase in average delay and the most decrease in network cost. The ratio of increase in average delay to the cost-savings of removed capacity is therefore used to select the appropriate action in each iteration. 3.3 Step 2. Rescheduling - Each link in isolation
The expansion generated by considering each period in isolation does not consider benefits of economies of scale. For example, it may be cheaper to install one expansion option of size X in period 1, than to put in say 0.5X in period 1 and 0.5X in period 3. A rescheduling step takes this into account. A simple illustration follows. Table 1 shows three expansion options for a hypothetical network. Notice the economy of scale. Table 2 shows the expansion plan obtained by considering each period in isolation, for two links in the network. Values in Table 2 are total installed capacities in each period. The total cost for expansion is {(18000 + 18000) + (10000 + 10000)} =
Total capacity (Kbps)
link 1 link 2
year 1
year 2
80 20
80 40
Table 3 shows a rescheduled plan where the second year expansion for link 1 has been brought forward to period 1. No change has been made to the schedule for link 2. The total cost of expansion has now reduced to {(29000) + (10000 + 10000)} = $49000. Rescheduling is done for each link independently, generating alternative schedules by moving some expansions forward in time, and then selecting the alternative with lowest cost. Note that rescheduling only involves bringing expansions forward in time. In this manner feasibility will be maintained. 3.4. Step 3. Rescheduling - Use excess capacity
Rescheduling of links individually presents one additional opportunity for cost reduction. Notice that ir T~ble 3, link 1 has excess capacity in year 1. It may now be possible to defer the year 1 expansion of link 2 or to eliminate it altogether if traffic on link 2 in year 1 could be rerouted to use excess capacity on link 1. Assuming this rerouting was feasible, the expansion plan will be as in Table 4. Operationally, the FDA is used to determine if such rerouting is feasible. The total expansion cost has reduced even further to {(29 000) + (18 000) } = $47 000.
$56 000.
Table 2 Expansion plan - Each period in isolation
Table 4 Rescheduled plan - Utilize excess capacity
Total capacity (Kbps)
link 1 link 2
401
year 1
year 2
40 20
80 40
Total capacity (Kbps)
link 1 link 2
year 1
year 2
80 0
80 40
402
A. Dutta, Y.K. Kim ~European Journalof OperationalResearch 91 (1996) 395-410
The three parameters which make the general expansion problem combinatorially explosive are the number of edges, capacity options and periods in the planning horizon. Our heuristic copes with this complexity by keeping one parameter fixed while looking at variations in others. Step 1, for instance, fixes the time period while making expansion decisions, while Step 2 fixes the edge when rescheduling over time. In a later section heuristic solutions are compared with a benchmark.
3.5. Algorithms
We now specify algorithms for individual components of the heuristic described informally in the previous section. These are then integrated into a complete heuristic for the capacity expansion problem. The notation introduced in Section 2 is used along with a few additional ones.
Additional Notation: C°t : Vector of capacities of all edges at start of period t in the final plan. Ci. : Vector of capacity at start of each period for link i in the final plan. DI o- : 1' in average delay when link j's capacity is decreased from Car to C i. Raij :(Ratio of delay I" to cost $) = Diij/(Parj - Pit ). Some variables will be defined locally within individual algorithms. Annotations appear in italics within [* . . . *] to associate algorithm steps with informal steps described earlier. The notation 'FDAQ {Demand, Capacity}' will denote application of the FDA to a given capacity assignment with a given traffic requirement. Recall that the FDA routes demand through the given capacity so as to minimize average delay. Procedure ONE-PERIOD-EXPAND corresponds to Step 1 above, while EOS-1 and EOS-2 correspond to Step- 2 and Step- 3 above where economy of scale is considered. For algorithmic convenience, capacity expansion options are assumed stored in order from largest to smallest.
P R O C E D U R E ONE-PERIOD-EXPAND (Input: Ceq, Ret2; Output: Coq): [* Expand Cot 1 to accommodate R o t 2 * ] FOR j = l t o S a D O ; Cj,~ <---Cj, 1 + d~r; [ * Begin by adding the maximum expansion option * ] Mark link j as not visited; END FOR; Atl = FDA Q {R°tl, C.tl} [* Apply FDA to augmented network and get delay * ] FOR j ' = l t o S a D O ; Let C/~--- C°t~ be modified such that Qtl = Cyq + di; [* The expansion on link j has been reduced from dar to d i *] A~ = FDA.. (~) {R°tl, C ° t ) . DIi~ = A'/1 - Ate; [* 1' in average delay due to capacity $ on link j * ] R a i / = D j y / ( P ~ - P/y); [ * Ratio of delay $ to cost $ *] END FOR; END FOR; DO WHILE Att < Amax; Let i', j' = arg rain { R a J ; IF link j' is marked not visited THEN mark link j' as visited Cy,t~ ~ Cy,q + di,; At~ *-- Aq + DIi,/; Rai,/<-ELSE Let i be the expansion option chosen when link j' was last visited. IF dj < d i THEN Cj, tl ~-- Cj,tl -~ dti; All ~ Atl -- nliT, + DIi,y,; ENDIF ENDIF ENDO
P R O C E D U R E EOS-1 (Input: Coq "'" Cotz; Output ~,q); [ * Take myopic expansion plan from period t 1 to t2 and reschedule the plan for each link individually. Output consists of rescheduled capacity as-
A. Dutta, Y.l( Kim / European Journal of Operational Research 91 (1996) 395-410
D O W H I L E A t < Amax; C# ~ Cjt - {Expansion on link j in C.t}; D I N C / ~ - oo; j' = arg min (DINCj); A t *-- A t + DINCy,; ENDDO; END EOS-2;
s i g n m e n t f o r period t 1 * ]
F O R j = to .Z~ DO; Generate alternate expansion schedules satisfying the condition: [ * See Fig. 4 f o r Details * ]
k=t I
403
k=t I
t x<~'~t 2
The mechanics of rescheduling the expansion plan for a link in EOS-1 is easier to see through Fig. 4. There, the contour {a, b, c, d, e, f, g, q} gives the expansion schedule for a link l, generated by E X P A N D - O N E - P E R I O D . Alternate schedules are generated by bringing these expansions forward in time. For instance, contours {a, b, c, d, h, g, q}, {a, b, k, e, f, g, q}, {a, b, c, k, e, h, g, q} and {a, p, n, k, j, h, g, q} are all alternate expansion schedules. Their costs will be determined by economy of scale benefits. The individual procedures shown above can now be integrated into the complete heuristic:
Select the least expensive expansion schedule for link j E N D FOR; END EOS-1; P R O C E D U R E EOS-2 (Input: ~ . , , R.t; Output: C.t); [* Utilize spare capacity in the o u t p u t o f E O S - 1 for period t * ] At = F D A Q { R • t , C•t}; = Define ~ t - - M o d i f i e d C . t wherein the expansion on link j in C . t is eliminated;
F O R J = 1 T O .~_ DO; PROCEDURE EXP-PLAN (Input: C•O, R s t , Amax, d i, Pij, H, 5¢, 5~; Output: A i , . . . , At_/, C • l , . . . , C.H); t~l;
DINCj = A~ - At; E N D FOR; Let j' = arg rain (DINC~); A t <-- A t + D I N C / ;
g
m
q
f ci • is the lowerenvelopeof
n
c
d
b
a
I 1
2
I
3 Periods
I
steps. Movingexpansionsforward in time gives alternateexpansion sehe~tule~, I I 4 5 6 --I1~
Fig. 4. Alternate expansion schedules for a link I.
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
404
W H I L E t < H DO;
k*--t; W H I L E k < H DO; A, = F D A (9 {R.k,Co,t_l}; IF A t _~
(9 {C.,,_ t,U.,t,U.,,+~,...,
8
130
6q
F
U.n};
C°, t = EOS - 2 (9 {C°, ,,R°,t}; t~t+l ENDDO; A n = F D A {R.H , C°,H_I}; IF A n _~
4. Computational results The heuristic E X P - P L A N was implemented on a VAX 6410, using F O R T R A N . It was tested using four networks with 5, 15, 30 and 50 nodes. These have 7, 21, 45 and 85 links respectively.
5°
200
Fig. 6. Fifteen node network.
They are shown in Figs. 5-8. A five year planning horizon was assumed. Twenty expansion options were used in the tests and their cost characteristics appear in Table 5. These options are constructed in a particular way. The first five are physically distinct options. The remaining fifteen are combinations of these. For example, the cost of 85 Kbps is the sum of the costs of 80 Kbps and 5 Kbps. Such combinations have been commonly used in the capacity expansion literature [8]. Computational experience will be presented in two parts. The first shows how computation time of the heuristic varies with different problem parameters. In the second part, expansion costs produced by the heuristic are compared to that obtained for a benchmark problem. These two sets of tests together allow one to determine whether the heuristic generates low cost expansion plans in a reasonable amount of time for problems of practical size.
4.1. Efficiency of heuristic
50 4 60 Fig. 5. Five node network.
The most time consuming operation in EXPP L A N is the FDA. The latter has a complexity of O(n2), n being the number of nodes [5]. Table 6 shows that the CPU time for EXP-PLAN is approximately proportional to n 2. This low order polynomial time behavior suggests that the heuristic could be used to develop expansion plans for reasonably large networks.
A. Dutta, Y.IC Kim / European Journal of Operational Research 91 (1996) 395-410
EXP-PLAN was applied to the 15 node problem (Fig. 6) for different annual traffic growth rates. As expected, Table 7 shows that the total expansion cost increases significantly with traffic growth rate. Note however, that execution time of the heuristic remains at around 15 seconds. Using the 15-node problem again, Table 8 shows the variation of total expansion cost and CPU time
with respect to Amax, the permissible average delay. Lower values of Amax require more capacity resulting in higher costs. Again, CPU time does not appear to vary significantly with Am~x. The preceding results indicate that the heuristic is quite fast. For additional support, the 50 node network (Fig. 8) was tested again to examine variation with respect to Areax and traffic
130
70
50
80
120
50
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30
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~
/
110
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60
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100
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100
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Fig. 7. T h i r t y n o d e n e t w o r k .
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A. Dutta, Y.K. Kim / EuropeanJournal of Operational Research 91 (1996) 395-410 Table 5 Cost structure of capacity expansion options Expansion options
Capacity (Kbits/sec)
Fixed cost ($)
Variable cost ($/km)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 10 20 40 80 85 90 100 120 160 165 170 180 200 240 245 250 260 280 320
5000 10000 20000 35000 50000 55000 60000 70000 85000 100000 105000 110000 120000 135000 150000 155000 160000 170000 185000 200000
50 80 140 210 300 350 380 440 510 600 650 680 740 810 900 950 980 1040 1110 1200
growth rate. T a b l e s 9 a n d 10 show the same b e h a v i o r of C P U time as was seen for the 15-node problem.
4.2. Developing a benchmark problem F o r c o m p a r i s o n , a b e n c h m a r k p r o b l e m def i n e d as follows. First, a p p r o x i m a t e the discrete cost f u n c t i o n by a continuous l i n e a r lower envelope. Next, k e e p r o u t i n g fixed across all time
Table 6 Relationship between network size and CPU time Network size
Average delay constraint
Networkcost (KS)
CPU time (sec)
5 15 30 50
0.04 sec 0.04sec 0.04sec 0.04 sec
124 1027 1507 3076
2.41 13.28 61.42 234.30
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Traffic demands between each node pair at beginning of horizon: 5 nodes: 25 pkt/sec. 15 nodes: 3.0 pkt/sec. 30 nodes: 0.7 pkt/sec. 50 nodes: 0.3 pkt/sec.
407
Table 7 Variation of cost and CPU time w.r.t, to traffic growth rate: 15 node network Annual traffic Growth rate (%)
Total cost (KS)
CPU time (sec)
5 10 15 20 25 30
567 765 1027 1248 1584 1995
11.63 12.46 13.28 16.28 20.96 17.87
Initial link capacity: 100 Kbps; Amax:0.04 sec. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
periods in the p l a n n i n g horizon; i.e. traffic bet w e e n a n O - D pair follows the same route(s) in each period. A n optimal s o l u t i o n to this m o d i f i e d p r o b l e m will not c o n s t i t u t e a lower b o u n d for o u r original one. However, in o u r view it is a good b e n c h m a r k against which to c o m p a r e costs generated by E X P - P L A N , for the following reasons. First, closed form solutions c a n b e easily comp u t e d for the m o d i f i e d p r o b l e m . Also, in the tests of Section 4.1, initial traffic a n d the rate of traffic increase per p e r i o d is the same for each O - D pair. U n d e r these c o n d i t i o n s a n d with c o n t i n u o u s capacity, capacity can b e i n c r e a s e d in just e n o u g h a m o u n t s to m e e t the delay c o n s t r a i n t Zimax. T h u s it is unlikely that r o u t i n g will have to be a l t e r e d
Table 8 Variation of expansion cost and CPU time w.r.t. Amax: 15 node network Delay limit '~max (sec)
Total cost
CPU time
(KS)
(sec)
0.025 0.030 0.035 0.040 0.045 0.050
1779 1391 1175 1027 921 791
11.06 11.43 11.94 13.28 15.01 14.24
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
A. Dutta, Y.K. Kdm~European Journalof OperationalResearch 91 (1996) 395-410
408
in different periods to divert traffic away from congested routes to routes whith excess capacity due to discreteness in the expansion options. Hence it appears reasonable to fix routing in the modified problem. So for this benchmark problem, we determine the routing in the starting period using FDA, and then keep it fixed for all remaining periods. Once again, we emphasize that the m o d i f i e d p r o b l e m is a b e n c h m a r k but not a lower bound. The benchmark problem can be formulated as: (BP) H
Minimize
~ •
S"
E
~it
(1)
t=li=l
subject to
~', i= 1 C i : - t + 2"it - f . _~
(2)
V i = 1, 2 . . . . , .W,
t = 1, 2 . . . . , H ,
(3)
C , = Ci,t_ l + :Wi, Vi = 1, 2 . . . . , .g~,
t = 1, 2 . . . . , H,
(4)
where: : Cost per unit capacity on link i using the continuous linear approximation. ~,t: Capacity (continuous) added on link i at the beginning of period t. 3', : Sum of packet rates for all O-D pairs in period t.
Table 9 CPU time for 50 node network: Traffic growth rate Annual traffic growth rate
Total cost (KS)
CPU time (sec)
5 10 15 20
951 1276 1746 2253
232.12 246.12 259.72 275.95
(%)
Delay limit
Total cost
CPU time
am~x(sec)
(KS)
(sec)
0.04 0.05 0.06
3076 1746 965
234.30 259.72 288.13
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 0.3 packet/sec.
Other notation is the same as defined previously. ~ , is continuous. (2) requires average delay to be less than Ama x in each period. Flow must not exceed capacity on any link (3). (4) connects link capacities in adjacent periods with the amount of the expansion. With fixed routing, fit = fi,t_l * ( l + ~ n t - l , t ) ,
Vt = 1, 2 . . . . . H,
Ci.t- 1 + A~it > fit
Table 10 C P U t i m e for 50 node network:Delay constraint
Initial link capacity: 100 Kbps; Amax:0.05 sec. Average packet size: 100 bits. Initial traffic between each node pair: 0.3 packet/sec
where ~Rt_l, t is the traffic growth rate from period t - 1 to t. (BP) then decomposes in to H subproblems, (BPt) one for each period. In view of (4), subproblems should be solved in increasing order of t. For a specific period z, the subproblem is: (BP~) Minimize
D = d i ~_~a,~iz
(5)
i=l
subject to A, = - "}lri= 1 Ci,r_l+~air-fir
<-~Amax'
C~,~_t + ~,+ > f i ~ V i = 1, 2 . . . . . .~.
(6) (7)
A global minimum can be found using Lagrange multipliers. Since ~-~. is continuous, inequality (6) can be replaced by an equality, and we will show that constraint (7) is redundant. Thus, the problem is minimizing (5) subject to (6). The closed form solution can be shown to be:
~it =fit + ~j=l d~jf~t "ytAmax
-
i = 1, 2 . . . . . .S".
J
fit ~ -- Ci,t_l, (12)
A. Dutta, Y.K. Kim / European Journal of OperationalResearch 91 (1996)395-410 Table 11 Unit costs of five expansion options for link length of 100 km Expansion option
capacity (Kbps)
Discrete unit cost a ($/Kbps)
Lower envelope ($/Kbps)
1 2 3 4 5
5 10 20 40 80
2000 1800 1700 1400 1000
1000 1000 1000 1000 1000
a Unit cost = (fixed cost + variable cost * 100 km)/capacity. Data of fixed and variable costs are given in Table 5. Since fit, di, 7t a n d d m a x have positive values, we have
gt~it + Ci,t-1 - f i t > O, satisfying c o n s t r a i n t (7). U s i n g Co0, the starting n e t w o r k capacity, expression (12) can b e used to c o m p u t e the e x p a n s i o n s of all successive periods.
4.3 Comparison between heuristic and benchmark solutions Note that the c o n t i n u o u s l i n e a r lower envelope is a relatively good a p p r o x i m a t i o n of the discrete capacity cost f u n c t i o n for large e x p a n s i o n o p t i o n s b u t n o t the smaller ones. T a b l e 11 illustrates this fact u s i n g cost data from T a b l e 5. T a b l e 12 a n d 13 c o m p a r e total costs o b t a i n e d by E X P - P L A N a n d the b e n c h m a r k u s i n g the 15-node p r o b l e m . T h e difference q u i t e low w h e n traffic growth rate is large o r Ama x is small. This is b e c a u s e in b o t h c i r c u m s t a n c e s large e x p a n s i o n Table 12 Comparison of heuristic and benchmark for different Amax Zlmax
Delay limit
PR c o s t (KS)
Benchmark cost ( K S )
Cost difference (%)
0.025 0.030 0.035 0.040 0.045 0.050
1779 1391 1175 1027 921 791
1741 1364 1096 894 738 612
2.2 2.0 7.2 14.9 24.8 29.2
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
409
Table 13 Comparison of heuristic and benchmark for different traffic growth rates Annual traffic growth rate (%)
PR cost (KS)
Benchmark (KS)
Cost difference
5 10 15 20 25 30
567 765 1027 1248 1584 1995
423 660 894 1170 1492 1854
34.0 15.9 14.9 6.7 6.2 7.6
Initial link capacity: 100 Kbps; Amax:0.04 sec. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 pkt/sec.
o p t i o n s are n e e d e d , for which the l i n e a r e n v e l o p e is a relatively good a p p r o x i m a t i o n . T h e cost diff e r e n c e is higher for large Amax or low traffic growth rates w h e n smaller sized e x p a n s i o n options are used. I n fact, T a b l e s 12 a n d 13 show differences a r o u n d 30%. W e believe that these i n s t a n c e s are c a u s e d by poor a p p r o x i m a t i o n of the lower e n v e l o p e r a t h e r t h a n poor p e r f o r m a n c e by E X P - P L A N . T a b l e 14 shows that the gap b e t w e e n E X P P L A N a n d b e n c h m a r k costs b e c o m e s smaller as the n u m b e r of n o d e s in the n e t w o r k increases. This suggests that the a p p r o x i m a t i o n of the heuristic may actually be i m p r o v i n g as the network gets larger. Actually, E X P - P L A N yielded slightly lower cost t h a n the b e n c h m a r k for o n e p r o b l e m i n s t a n c e ( u n d e r l i n e d in T a b l e 14). O n e Table 14 Comparison of heuristic and benchmark for varying network size Network size
PR c o s t (KS)
Benchmark cost ( K S )
Cost difference
5 15 30 50
124 1027 1507 3076
82 894 1357 3112
51.2 14.9 11.1 -0.1
Initial link capacity: 100 Kbps; Amax:0.04 sec. Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 5 nodes: 25 pkt/sec. 15 nodes: 3.0 pkt/sec. 30 nodes: 0.7 pkt/sec. 50 nodes: 0.3 pkt/sec.
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reason for this occurrence lies in i m p l e m e n t a t i o n o f the F D A . Even t h o u g h the F D A theoretically finds an optimal routing, its gradient n a t u r e makes c o n v e r g e n c e slow. F o r this reason most i m p l e m e n t a t i o n s stop searching w h e n i m p r o v e m e n t in average delay is less than a specified ' t o l e r a n c e ' [5]. T h e F D A is used by the b e n c h m a r k as well as E X P - P L A N , and the ' t o l e r a n c e ' m a y occasionally p r o d u c e such an u n e x p e c t e d result. H o w e v e r these occasions arose very infrequently during testing. W h e n they do, it is just an indication that the heuristic and b e n c h m a r k solutions are so close that the tolerance in F D A ' s termination condition can give rise to such anomalies.
5. Conclusion This p a p e r suggests a heuristic a p p r o a c h to multiperiod network capacity expansion, which consists o f first determining a myopic expansion plan based on single period heuristics, which is then subsequently rescheduled for e c o n o m i e s of scale effects. Several variations o f this basic idea can be implemented. F o r instance, an integer p r o g r a m m i n g formulation like [6] m a y be used to d e t e r m i n e the expansion plan of each period in isolation. Alternatively the F D A m a y be used in conjunction with cut-saturation heuristics [8] for the same purpose. Some caution must be exercised in using the b e n c h m a r k to judge solution quality of E X P - P L A N . As observed previously, the linear lower envelope will be a p o o r approxim a t i o n for those specific instances in which the cost structure is sharply ' b o w e d ' . T h e r e f o r e we suggest that in these situations, it may m o r e practical to c o m p a r e the heuristic solution against several b e n c h m a r k s obtained using continuous linear approximations that lie within a lower and u p p e r envelope o f the actual discrete cost function.
Acknowledgment T h e authors wish to t h a n k a n o n y m o u s refe r e e # 2 whose detailed c o m m e n t s were very helpful in revising the paper.
References [1] Bell, T.E., "Telecommunications", in: Technology 1992, IEEE Spectrum, January (1992) 36-38. [2] Boorstyn, R.R., and Frank, H., "Large-scale network topology optimization", IEEE Transactions on Communications 25/1 (1977) 29-46. [3] Dutta, A., and Lim, J.I., "Multiperiod capacity planning for packet-switched telecommunication networks", Operations Research 40/4 (1992) 689-705. [4] Frank H., and Chou, W., "Topological optimization of computer networks", Proceedings of the IEEE 60/11 (1972) 1386-1397. [5] Fratta, L., Gerla, M., and Kleinrock, L., "The flow deviation method: An approach to store-and-forward computer communication network design", Networks 3 (1973) 97-133. [6] Gavish, B., and Neuman, I., "A system for routing and capacity assignment in computer communication networks", IEEE Transactions on Communications 37/4 (1989) 360-366. [7] Gerla, M., "The design of store-and-forward networks for computer communications", Ph.D. dissertation, School of Eng. and Appl Sci., Univ. of California, Los Angeles, CA, 1973, [8] Gerla, M., and Kleinrock, L., "On the topological design of distributed computer networks", IEEE Transactions on Communications 25/1 (1977) 48-60. [9] Kleinrock, L., Queuing Systems: Volume 11. Computer Applications, Wiley-Interscience, New York, 1976. [10] Luss, H., "Operations Research and capacity expansion problems: A survey", Operations Research 30/5 (1982) 907-947. [11] Monma, C.L., and Sheng, D.D., "Backbone network design and performance analysis: A methodology for packet switching networks", IEEE Journal on Selected Areas in Communication 4 (1986) 946-965. [12] Ng, T.M.J., and Hoang, D.B., "Joint optimization of capacity and flow assignment in a packet-switched communications network", IEEE Transactions on Communications 35/2 (1987) 202-209. [13] Papadimitriou, C.H., "The complexity of the capacitated tree problem", Networks 8 (1978) 217-230. [14] Schwartz, M., and Stern, T., "Routing techniques used in computer communication networks", IEEE Transactions on Communications 28/4 (1980) 539-552. [15] Smith, R.L., "Deferral strategies for a dynamic communications networks", Networks 9 (1979) 61-87. [16] Yaged, B., "Minimum cost routing for dynamic network models", Networks 3 (1973) 193-224. [17] Zadeh, N., "On building minimum cost communications network over time", Networks 4 (1974) 19-34. [18] Zerbiec, T.G., "Considering the past and anticipating the future for private data networks", IEEE Communications, March (1992) 36-46.
European Journal of Operational Research 91 (1996) 395-410
Theory and Methodology
A heuristic approach for capacity expansion of packet networks Amitava
Dutta
a,., Young
Ki Kim b,..
a Department of Decision Sciences and MIS, College of Business, George Mason University, 4400 University Drive, Fairfax, VA 22030-4444, USA b Department of Management Science, College of Business, University oflowa, Iowa City, IA 52242, USA
Received March 1994; revised February 1995
Abstract
We address the problem of expanding transmission capacity of an existing packet network over a multiperiod planning horizon, the objective being low total cost of expansion. Discrete capacity choices, interaction with routing decisions, and economy of scale in the cost of capacity make it extremely difficult to decide when, where and how much capacity to add. A fast heuristic solution method is developed based on the well established Flow Deviation routing algorithm. The heuristic begins by making myopic expansion decisions, which are then subsequently adjusted to account for economies of scale in the cost of capacity. Heuristic solutions are compared to a benchmark which approximates the real cost function by its linear lower envelope. Since the number of possible expansion plans is an exponential function of the number of edges, capacity choices, and periods in the planning horizon, a fast heuristic allows one to look beyond small problems at more realistically sized ones. Keywords: Heuristics; Communication; Capacity expansion
1. Introduction
The d e m a n d for telecommunications services has exploded in recent times. Technological advances have m a d e it cheaper to communicate electronically. The decreasing cost of computing power has also led to the distribution of computing activities and a need to communicate among different computing agents. While voice has been and still continues to be the dominant type of traffic, the importance of data applications is on the rise. One prediction estimates that data and * Corresponding author. Author's current address: SongPa-Gu, GaRak-Dong, 1915 Seoul, South Korea. **
voice will constitute 61% and 39% of private network traffic respectively, by 1995 [18]. Applications such as electronic mail, C A D / C A M , order entry, financial and accounting inquiries have contributed to this increase in data volume. We will not elaborate further on these trends as they are widely recognized. For some recent examples of trends see [1]. In this paper, we are interested in one important consequence, which is the need to expand network transmission capacity in response to anticipated increase in demand. Since capacity is expensive, cost minimization is an important objective in generating expansion plans. Consider a hypothetical wide area network shown in Fig. 1. The nodes represent origins/destinations for traffic. Over time, the traffic demands of
0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00089-5
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A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
existing origin-destination (O-D) pairs may increase a n d / o r new nodes may be added to the network, requiring additional capacity. Fig. 2 shows the cost structure of transmission capacity, the two characteristics being (i) discreteness and (ii) economy of scale. The latter implies that it may be beneficial to install large expansion options to meet future demand, rather than several small options as needed [10]. Thus a multiperiod view of the expansion problem is needed. Fig. 1 also shows that capacity and routing decisions will be closely related, and there are many alternatives for the latter. These characteristics make the expansion problem an intractable one. A single expansion plan consists of a capacity assignment for each edge of the network for each period. Even for a small network of 10 nodes, 20 edges, 4 capacity choices and a 5 period planning horizon, there will be approximately 10 60 expansion plans. In fact the problem remains intractable for H = 1, since this single period problem is a more general case of the capacitated minimum spanning tree, which is known to be NP-hard [13].
1.1. Related literature A large number of heuristics exist for single period capacity assignment. They cope with complexity by iterating between capacity and routing decisions until some appropriate stopping criteria are met [2,4,7,8,11,12]. These heuristics are quite
+130 140 ~
200 Fig. 1. A widearea network.
~. 100
C 0 $
t
I Capacity Fig. 2. Generalcost structureof capacity.
fast and their effectiveness has been established through extensive testing with many and varied problem instances, rather than through comparison with lower bounds. The integer programming model in [6] is an exception in that it jointly optimizes routing and capacity and also produces a lower bound for comparison. Computation time is not reported in [6]. Literature on the multiperiod expansion problem addressed in this paper is somewhat more scarce. While heuristic approaches also predominate here, considerable simplifying assumptions have been made. For instance, the cost function was assumed to be a concave function of link capacity in [16]. In [17] traffic for communicating node pairs was routed over the same path for all time periods, ignoring the interplay between capacity and routing. The 'deferral' strategy proposed in [15] postpones installation of additional capacity for as long as possible by routing traffic around saturated links, thereby ignoring economy of scale benefits. The integer programming formulation in [3] does not make such simplifying assumptions. However, while solution quality is good, computation time increases sharply with problem size. In short, we have efficient heuristics for capacity expansion which make major simplifications of problem characteristics, and optimization models which are more realistic but can only be solved approximately, and with considerable computational effort. In this paper we develop and test a heuristic that can produce low cost expansion plans for large problem instances within reasonable computation times, without having to make gross
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
397
simplifications about major problem characteristics such as the cost structure of capacity or routing policies. While a lower bound cannot be conveniently computed, a benchmark problem is described to judge quality of the heuristic solutions.
Objective:
2. Problem statement and characteristics
2.1. Cost of capacity and timing of expansions
We now give a more precise statement of the expansion problem and assumptions about different components thereof. Major characteristics, such as discrete cost, have not been simplified to make the problem tractible. However, as with any abstraction of a physical problem, some approximations by omission and commission have been made in the problem statement. Their impact on the applicability of the heuristic solution method is discussed.
Note that we start with existing link capacities rather than assume links to have zero initial capacity. In practice, one often does not have the opportunity to plan capacity expansions from scratch. We assume that expansion decisions are made periodically rather than continuously. This is quite realistic in most circumstances. However, we also assume that the extra capacity for a period is installed instantaneously at the start of the period. Although this assumption is not quite realistic, and results in some cost overestimation, it is commonly made in the capacity expansion literature for reasons of tractability. The overestimation decreases as periods become shorter relative to the length of the planning horizon. We do not make any simplifying assumptions about discreteness of capacity or the presence of economies of scale. Typically, costs of capacity have fixed and variable components. The former is dependent on the type and size of link being installed, while the latter is usually tied to the physical distance covered. Therefore we retain the essential elements of cost structure in our problem.
Notation: : Number of periods in planning horizon. : Number of links in the network. : Traffic demand for O-D pair s in period t. Rst A max : Maximum performance limit on average delay. : Computed average network delay in peA, riod t. : Number of capacity expansion options. : Capacity of expansion option i, i = di 0,1 . . . . . ,7,(; d o = 0. P,j : Cost of installing expansion option i on link j : Total capacity on link i at start of period t Cit in the final plan. L, : Flow assigned to link i in period t. H
The capacity expansion problem can now be stated as follows: Given:
{ H,
Cio Vi, Rst Vs, t, di, i = 1 . . . . . ~e';
Pij, i = 1 . . . . . ,.,2r; j = 1 . . . . . c~}. Decide:
{Ci, and fit, i = 1,...,.o~; t = 1 . . . . . H } .
.~
Min
H
Y'~ ~ ( C i t - C
i,t_l)
i=1 t=l
subject to At
_~
2.2. Routing method Due to the relationship between capacity and routing decisions both are produced as outputs of our solution method. In particular, the route(s) assigned to the traffic for any specific O-D pair can change from period to period. However, many routing methods exist [14], and we have to make some choice among them. Dynamic routing methods respond to short term variations in network traffic by altering the paths used, while static methods do not. However, static routing is easy to describe and allows the direct computation of average delay for a given network capacity. Also,
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A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
it has been observed that at steady state, flow patterns and delays induced by good dynamic routing policies are very close to those obtained with optimal static routing [8]. It has therefore been common in the literature to employ static routing for capacity planning problems and we do so here as well. Two major variations of static routing are the bifurcated and nonbifurcated cases. Take node pair (1, 11) in Fig. 1 for example. If bifurcated routing were used, then in a given period, packets between 1 and 11 could be distributed over three paths say {1, 3, 12, 5, 11}, {1, 2, 4, 5, 11} and {1, 3, 4, 5, 11}. If nonbifurcated routing were employed, all packets between 1 and 11 would have to follow one path (in a period), say {1, 3, 4, 5, 11}. We use nonbifurcated routing as it tends to better utilize available network capacity. Other static routing algorithms could be easily substituted in its place within our heuristic.
2.3. Performance criterion The common performance criterion for a data network is average packet delay. Each link in the network is a server whose service rate is determined by its capacity, while packets routed on that link represent customers. This produces a network of queues and, under some assumptions that have been found to be quite robust, the average packet delay in period t, At, is commonly expressed as [9]
At = -3"i=1
fit "fit
(A t ~ oo if Cit
2.4. Limitations of problem statement The effect of assuming instantaneous capacity installation has already been discussed. In addition, the problem statement implicitly assumes that the number of nodes and topology of the network remain constant over time. This is not a serious limitation. New nodes introduced in future periods may be included from the start and assigned a traffic demand of zero for all periods in which they physically do not exist. Similar conditioning of data can be used to handle changes in topology where the number of nodes is unchanged but new links are added between existing nodes. Note also, that the problem statement implicitly assumes the nominal cost structure of capacity to remain fixed over time. This has no effect on the algorithm for our heuristic and has been made solely for implementation convenience to avoid having to store several arrays of costs, one for each period. If separate cost arrays are used for each period, it is possible to capture changes in nominal cost. However, these storage requirements can quickly become quite considerable. One assumption not readily apparent, is that capacity, once installed, is not removed for the remainder of the planning horizon. Physical removal is usually necessitated by aging or technical obsolescence and occurs far less frequently than the installation of additional capacity. Thus, for any given instance of the expansion problem, very few links would be candidates for removal. These links can be assigned a capacity of zero at the start of the appropriate period, and the heuristic can then assign a new capacity to them. In summary, the heuristic to be developed can, with some preconditioning of input parameters, be applied to several variations of the capacity expansion problem.
3. Solution method
Since the problem is NP-hard, we aim for good approximate solutions. Also, since we want reasonably quick approximations for large problems,
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
it is natural to see if efficient algorithms for single period capacity assignment, referenced in Section 1.1, can be adapted for the multiperiod problem.
Co
RI
Note that a good expansion plan cannot be obtained simply by applying existing single period heuristics once for each period in the planning
R,
R2
Period~
Period2
Periods
PeriOd4
Period5
Period6
6) Treat each perid in isolation. For each period i, use a single period heuristic to increase capacity from C Oto a vahie~ i which accommodates Ri while staying just below the allowed delay. Resulting sequence ~ . ."~6, is myopic since it does not consider economies of scale.
E1
~2
~3
C4
~5
C6
(ii)
From the sequence ~ - l . . . ~6. extract the six expansion values for any one link. Resehedule these six expansions considering economy of scale benefits. Repeat for each link in the network. Retain only the rescheduled assignments of all links for period 1, giving vector ~ .
col I
R2
R3
R5
R~
(iii)
In vector ~l, eliminate small expansion choices from selected links. This yields final capacity expansion decision for period 1, giving vector C 1.
colR i C1
R2
R3
C2 . .. C6 to be determined in like manner (iv)
Repeat steps (ii) through (iv) for i=2,...,6, using
399
Ci.I as the starting capacity.
Fig. 3. Heuristic for capacity expansion.
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horizon. Such a procedure would ignore tradeoffs involving economies of scale in the cost of capacity. Further, single period heuristics generally assume zero initial capacities on links. Substantial modifications would be necessary to ensure that the capacity at the end of one period is the starting point for expansion in the next period. In short, adaptation from single to the multiperiod case involves several intricacies. 3.1 Overview of heuristic
In Fig. 3, we first present an informal overview of the major components of our heuristic. Algorithmic specification will appear subsequently. Fig. 3(i) shows some parameters of the expansion problem on a time line. Ri is the vector of traffic requirements of all O-D pairs in period i, while C O is the vector of capacities on different edges of the network at the end of period 0. We need to determine C1 . . . . . C6, the capacity vectors at the start of periods 1 through 6. The final capacity decision for each period i, Ci, is reached in three steps. First an initial expansion plan is generated where demand for each period is met independent of that in others. This is done using existing single period algorithms which are known to be fast. This myopic plan is then modified in the second step by rescheduling the expansion plan of each link independent of other links, in order to use economies of scale. This step can be viewed as being myopic in space since each link is treated in isolation. The third step attempts to utilize excess capacity installed on some links due to rescheduling, by eliminating small expansions on some links and rerouting their traffic through this spare capacity. This results in further cost reductions. Details of each step follow. 3.2. Step 1. Expansion - Each period in isolation
The basic operation consists of expanding Ci to accommodate the demand of some future period R j, j > i. To do so, Rj must first be routed 'judiciously' through C i to minimize average de-
lay. There can be three outcomes. The average delay may be less than the permissible limit, more than the limit, or it may be simply infeasible to route Rj through C i with finite delay. The last two outcomes will require expansion of capacities on an 'appropriately' selected collection of links. The Flow Deviation Algorithm (FDA) [5] is among the best established ones to judiciously route traffic demand through a given network capacity. Since details may be found in [5], we only summarize the workings of the FDA to motivate its use in our heuristic. Capacity and traffic requirements are given as inputs to the FDA. It starts from an initial feasible flow and attempts to iteratively adjust this flow in a steepest-descent manner, so as to reduce average delay, until no significant reduction in delay is possible. The rate of descent is measured by the ratio of change of delay with respect to change in flOW, (OA/~fi) , for each link i. The FDA is summarized as follows:
P R O C E D U R E FDA; Step 1. Find a feasible starting flow f 0 and set n~-0 Step 2. f n + l ~ (1 - h ) f n + hi7~ Step 3. IF ( A ( f ~) - A(fn+l)) < 0 STOP; ELSE n~n+l and go to Step 2; where A ( f ~) is average network delay under flow fin, v n is shortest route flow under the metric l k = (OA/Ofk) , A minimizes A((1 - A ) f n + AtTn), and 0 is an acceptable tolerance. END FDA;
The process of expanding C i to accommodate Rj begins by temporarily adding the largest and most expensive capacity option to each link in C i. FDA is used to route Rj through this network. (We are assuming that the largest available expansion option, when added to Ci, will be sufficient to meet Rj within the permissible delay limit. Otherwise the entire problem is infeasible.) The average delay for this overcapacitated network will probably be much lower than the permissible limit. So we iteratively choose smaller
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410 Table 3 Rescheduled plan - Adjustment for link 1
Table 1 Example expansion costs Expansion options
Capacity (Kbits/sec)
Cost (KS)
1
20 40 80
10 18 29
2 3
capacity expansions on selected links, until average delay increases to a point just under the permissible limit. Using a greedy approach, we would like each iteration to result in the least increase in average delay and the most decrease in network cost. The ratio of increase in average delay to the cost-savings of removed capacity is therefore used to select the appropriate action in each iteration. 3.3 Step 2. Rescheduling - Each link in isolation
The expansion generated by considering each period in isolation does not consider benefits of economies of scale. For example, it may be cheaper to install one expansion option of size X in period 1, than to put in say 0.5X in period 1 and 0.5X in period 3. A rescheduling step takes this into account. A simple illustration follows. Table 1 shows three expansion options for a hypothetical network. Notice the economy of scale. Table 2 shows the expansion plan obtained by considering each period in isolation, for two links in the network. Values in Table 2 are total installed capacities in each period. The total cost for expansion is {(18000 + 18000) + (10000 + 10000)} =
Total capacity (Kbps)
link 1 link 2
year 1
year 2
80 20
80 40
Table 3 shows a rescheduled plan where the second year expansion for link 1 has been brought forward to period 1. No change has been made to the schedule for link 2. The total cost of expansion has now reduced to {(29000) + (10000 + 10000)} = $49000. Rescheduling is done for each link independently, generating alternative schedules by moving some expansions forward in time, and then selecting the alternative with lowest cost. Note that rescheduling only involves bringing expansions forward in time. In this manner feasibility will be maintained. 3.4. Step 3. Rescheduling - Use excess capacity
Rescheduling of links individually presents one additional opportunity for cost reduction. Notice that ir T~ble 3, link 1 has excess capacity in year 1. It may now be possible to defer the year 1 expansion of link 2 or to eliminate it altogether if traffic on link 2 in year 1 could be rerouted to use excess capacity on link 1. Assuming this rerouting was feasible, the expansion plan will be as in Table 4. Operationally, the FDA is used to determine if such rerouting is feasible. The total expansion cost has reduced even further to {(29 000) + (18 000) } = $47 000.
$56 000.
Table 2 Expansion plan - Each period in isolation
Table 4 Rescheduled plan - Utilize excess capacity
Total capacity (Kbps)
link 1 link 2
401
year 1
year 2
40 20
80 40
Total capacity (Kbps)
link 1 link 2
year 1
year 2
80 0
80 40
402
A. Dutta, Y.K. Kim ~European Journalof OperationalResearch 91 (1996) 395-410
The three parameters which make the general expansion problem combinatorially explosive are the number of edges, capacity options and periods in the planning horizon. Our heuristic copes with this complexity by keeping one parameter fixed while looking at variations in others. Step 1, for instance, fixes the time period while making expansion decisions, while Step 2 fixes the edge when rescheduling over time. In a later section heuristic solutions are compared with a benchmark.
3.5. Algorithms
We now specify algorithms for individual components of the heuristic described informally in the previous section. These are then integrated into a complete heuristic for the capacity expansion problem. The notation introduced in Section 2 is used along with a few additional ones.
Additional Notation: C°t : Vector of capacities of all edges at start of period t in the final plan. Ci. : Vector of capacity at start of each period for link i in the final plan. DI o- : 1' in average delay when link j's capacity is decreased from Car to C i. Raij :(Ratio of delay I" to cost $) = Diij/(Parj - Pit ). Some variables will be defined locally within individual algorithms. Annotations appear in italics within [* . . . *] to associate algorithm steps with informal steps described earlier. The notation 'FDAQ {Demand, Capacity}' will denote application of the FDA to a given capacity assignment with a given traffic requirement. Recall that the FDA routes demand through the given capacity so as to minimize average delay. Procedure ONE-PERIOD-EXPAND corresponds to Step 1 above, while EOS-1 and EOS-2 correspond to Step- 2 and Step- 3 above where economy of scale is considered. For algorithmic convenience, capacity expansion options are assumed stored in order from largest to smallest.
P R O C E D U R E ONE-PERIOD-EXPAND (Input: Ceq, Ret2; Output: Coq): [* Expand Cot 1 to accommodate R o t 2 * ] FOR j = l t o S a D O ; Cj,~ <---Cj, 1 + d~r; [ * Begin by adding the maximum expansion option * ] Mark link j as not visited; END FOR; Atl = FDA Q {R°tl, C.tl} [* Apply FDA to augmented network and get delay * ] FOR j ' = l t o S a D O ; Let C/~--- C°t~ be modified such that Qtl = Cyq + di; [* The expansion on link j has been reduced from dar to d i *] A~ = FDA.. (~) {R°tl, C ° t ) . DIi~ = A'/1 - Ate; [* 1' in average delay due to capacity $ on link j * ] R a i / = D j y / ( P ~ - P/y); [ * Ratio of delay $ to cost $ *] END FOR; END FOR; DO WHILE Att < Amax; Let i', j' = arg rain { R a J ; IF link j' is marked not visited THEN mark link j' as visited Cy,t~ ~ Cy,q + di,; At~ *-- Aq + DIi,/; Rai,/<-ELSE Let i be the expansion option chosen when link j' was last visited. IF dj < d i THEN Cj, tl ~-- Cj,tl -~ dti; All ~ Atl -- nliT, + DIi,y,; ENDIF ENDIF ENDO
P R O C E D U R E EOS-1 (Input: Coq "'" Cotz; Output ~,q); [ * Take myopic expansion plan from period t 1 to t2 and reschedule the plan for each link individually. Output consists of rescheduled capacity as-
A. Dutta, Y.l( Kim / European Journal of Operational Research 91 (1996) 395-410
D O W H I L E A t < Amax; C# ~ Cjt - {Expansion on link j in C.t}; D I N C / ~ - oo; j' = arg min (DINCj); A t *-- A t + DINCy,; ENDDO; END EOS-2;
s i g n m e n t f o r period t 1 * ]
F O R j = to .Z~ DO; Generate alternate expansion schedules satisfying the condition: [ * See Fig. 4 f o r Details * ]
k=t I
403
k=t I
t x<~'~t 2
The mechanics of rescheduling the expansion plan for a link in EOS-1 is easier to see through Fig. 4. There, the contour {a, b, c, d, e, f, g, q} gives the expansion schedule for a link l, generated by E X P A N D - O N E - P E R I O D . Alternate schedules are generated by bringing these expansions forward in time. For instance, contours {a, b, c, d, h, g, q}, {a, b, k, e, f, g, q}, {a, b, c, k, e, h, g, q} and {a, p, n, k, j, h, g, q} are all alternate expansion schedules. Their costs will be determined by economy of scale benefits. The individual procedures shown above can now be integrated into the complete heuristic:
Select the least expensive expansion schedule for link j E N D FOR; END EOS-1; P R O C E D U R E EOS-2 (Input: ~ . , , R.t; Output: C.t); [* Utilize spare capacity in the o u t p u t o f E O S - 1 for period t * ] At = F D A Q { R • t , C•t}; = Define ~ t - - M o d i f i e d C . t wherein the expansion on link j in C . t is eliminated;
F O R J = 1 T O .~_ DO; PROCEDURE EXP-PLAN (Input: C•O, R s t , Amax, d i, Pij, H, 5¢, 5~; Output: A i , . . . , At_/, C • l , . . . , C.H); t~l;
DINCj = A~ - At; E N D FOR; Let j' = arg rain (DINC~); A t <-- A t + D I N C / ;
g
m
q
f ci • is the lowerenvelopeof
n
c
d
b
a
I 1
2
I
3 Periods
I
steps. Movingexpansionsforward in time gives alternateexpansion sehe~tule~, I I 4 5 6 --I1~
Fig. 4. Alternate expansion schedules for a link I.
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
404
W H I L E t < H DO;
k*--t; W H I L E k < H DO; A, = F D A (9 {R.k,Co,t_l}; IF A t _~
(9 {C.,,_ t,U.,t,U.,,+~,...,
8
130
6q
F
U.n};
C°, t = EOS - 2 (9 {C°, ,,R°,t}; t~t+l ENDDO; A n = F D A {R.H , C°,H_I}; IF A n _~
4. Computational results The heuristic E X P - P L A N was implemented on a VAX 6410, using F O R T R A N . It was tested using four networks with 5, 15, 30 and 50 nodes. These have 7, 21, 45 and 85 links respectively.
5°
200
Fig. 6. Fifteen node network.
They are shown in Figs. 5-8. A five year planning horizon was assumed. Twenty expansion options were used in the tests and their cost characteristics appear in Table 5. These options are constructed in a particular way. The first five are physically distinct options. The remaining fifteen are combinations of these. For example, the cost of 85 Kbps is the sum of the costs of 80 Kbps and 5 Kbps. Such combinations have been commonly used in the capacity expansion literature [8]. Computational experience will be presented in two parts. The first shows how computation time of the heuristic varies with different problem parameters. In the second part, expansion costs produced by the heuristic are compared to that obtained for a benchmark problem. These two sets of tests together allow one to determine whether the heuristic generates low cost expansion plans in a reasonable amount of time for problems of practical size.
4.1. Efficiency of heuristic
50 4 60 Fig. 5. Five node network.
The most time consuming operation in EXPP L A N is the FDA. The latter has a complexity of O(n2), n being the number of nodes [5]. Table 6 shows that the CPU time for EXP-PLAN is approximately proportional to n 2. This low order polynomial time behavior suggests that the heuristic could be used to develop expansion plans for reasonably large networks.
A. Dutta, Y.IC Kim / European Journal of Operational Research 91 (1996) 395-410
EXP-PLAN was applied to the 15 node problem (Fig. 6) for different annual traffic growth rates. As expected, Table 7 shows that the total expansion cost increases significantly with traffic growth rate. Note however, that execution time of the heuristic remains at around 15 seconds. Using the 15-node problem again, Table 8 shows the variation of total expansion cost and CPU time
with respect to Amax, the permissible average delay. Lower values of Amax require more capacity resulting in higher costs. Again, CPU time does not appear to vary significantly with Am~x. The preceding results indicate that the heuristic is quite fast. For additional support, the 50 node network (Fig. 8) was tested again to examine variation with respect to Areax and traffic
130
70
50
80
120
50
60
30
,
~
/
110
I00
60
100
40
150
405
180
120
70
80
100
ll0
50
100
250
100
130
180
150
160
I00
I00
Ioo
oo
140
60
170
Fig. 7. T h i r t y n o d e n e t w o r k .
90
§
O
0
o~
I
o~
~
A. Dutta, Y.K. Kim / EuropeanJournal of Operational Research 91 (1996) 395-410 Table 5 Cost structure of capacity expansion options Expansion options
Capacity (Kbits/sec)
Fixed cost ($)
Variable cost ($/km)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5 10 20 40 80 85 90 100 120 160 165 170 180 200 240 245 250 260 280 320
5000 10000 20000 35000 50000 55000 60000 70000 85000 100000 105000 110000 120000 135000 150000 155000 160000 170000 185000 200000
50 80 140 210 300 350 380 440 510 600 650 680 740 810 900 950 980 1040 1110 1200
growth rate. T a b l e s 9 a n d 10 show the same b e h a v i o r of C P U time as was seen for the 15-node problem.
4.2. Developing a benchmark problem F o r c o m p a r i s o n , a b e n c h m a r k p r o b l e m def i n e d as follows. First, a p p r o x i m a t e the discrete cost f u n c t i o n by a continuous l i n e a r lower envelope. Next, k e e p r o u t i n g fixed across all time
Table 6 Relationship between network size and CPU time Network size
Average delay constraint
Networkcost (KS)
CPU time (sec)
5 15 30 50
0.04 sec 0.04sec 0.04sec 0.04 sec
124 1027 1507 3076
2.41 13.28 61.42 234.30
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Traffic demands between each node pair at beginning of horizon: 5 nodes: 25 pkt/sec. 15 nodes: 3.0 pkt/sec. 30 nodes: 0.7 pkt/sec. 50 nodes: 0.3 pkt/sec.
407
Table 7 Variation of cost and CPU time w.r.t, to traffic growth rate: 15 node network Annual traffic Growth rate (%)
Total cost (KS)
CPU time (sec)
5 10 15 20 25 30
567 765 1027 1248 1584 1995
11.63 12.46 13.28 16.28 20.96 17.87
Initial link capacity: 100 Kbps; Amax:0.04 sec. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
periods in the p l a n n i n g horizon; i.e. traffic bet w e e n a n O - D pair follows the same route(s) in each period. A n optimal s o l u t i o n to this m o d i f i e d p r o b l e m will not c o n s t i t u t e a lower b o u n d for o u r original one. However, in o u r view it is a good b e n c h m a r k against which to c o m p a r e costs generated by E X P - P L A N , for the following reasons. First, closed form solutions c a n b e easily comp u t e d for the m o d i f i e d p r o b l e m . Also, in the tests of Section 4.1, initial traffic a n d the rate of traffic increase per p e r i o d is the same for each O - D pair. U n d e r these c o n d i t i o n s a n d with c o n t i n u o u s capacity, capacity can b e i n c r e a s e d in just e n o u g h a m o u n t s to m e e t the delay c o n s t r a i n t Zimax. T h u s it is unlikely that r o u t i n g will have to be a l t e r e d
Table 8 Variation of expansion cost and CPU time w.r.t. Amax: 15 node network Delay limit '~max (sec)
Total cost
CPU time
(KS)
(sec)
0.025 0.030 0.035 0.040 0.045 0.050
1779 1391 1175 1027 921 791
11.06 11.43 11.94 13.28 15.01 14.24
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
A. Dutta, Y.K. Kdm~European Journalof OperationalResearch 91 (1996) 395-410
408
in different periods to divert traffic away from congested routes to routes whith excess capacity due to discreteness in the expansion options. Hence it appears reasonable to fix routing in the modified problem. So for this benchmark problem, we determine the routing in the starting period using FDA, and then keep it fixed for all remaining periods. Once again, we emphasize that the m o d i f i e d p r o b l e m is a b e n c h m a r k but not a lower bound. The benchmark problem can be formulated as: (BP) H
Minimize
~ •
S"
E
~it
(1)
t=li=l
subject to
~', i= 1 C i : - t + 2"it - f . _~
(2)
V i = 1, 2 . . . . , .W,
t = 1, 2 . . . . , H ,
(3)
C , = Ci,t_ l + :Wi, Vi = 1, 2 . . . . , .g~,
t = 1, 2 . . . . , H,
(4)
where: : Cost per unit capacity on link i using the continuous linear approximation. ~,t: Capacity (continuous) added on link i at the beginning of period t. 3', : Sum of packet rates for all O-D pairs in period t.
Table 9 CPU time for 50 node network: Traffic growth rate Annual traffic growth rate
Total cost (KS)
CPU time (sec)
5 10 15 20
951 1276 1746 2253
232.12 246.12 259.72 275.95
(%)
Delay limit
Total cost
CPU time
am~x(sec)
(KS)
(sec)
0.04 0.05 0.06
3076 1746 965
234.30 259.72 288.13
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 0.3 packet/sec.
Other notation is the same as defined previously. ~ , is continuous. (2) requires average delay to be less than Ama x in each period. Flow must not exceed capacity on any link (3). (4) connects link capacities in adjacent periods with the amount of the expansion. With fixed routing, fit = fi,t_l * ( l + ~ n t - l , t ) ,
Vt = 1, 2 . . . . . H,
Ci.t- 1 + A~it > fit
Table 10 C P U t i m e for 50 node network:Delay constraint
Initial link capacity: 100 Kbps; Amax:0.05 sec. Average packet size: 100 bits. Initial traffic between each node pair: 0.3 packet/sec
where ~Rt_l, t is the traffic growth rate from period t - 1 to t. (BP) then decomposes in to H subproblems, (BPt) one for each period. In view of (4), subproblems should be solved in increasing order of t. For a specific period z, the subproblem is: (BP~) Minimize
D = d i ~_~a,~iz
(5)
i=l
subject to A, = - "}lri= 1 Ci,r_l+~air-fir
<-~Amax'
C~,~_t + ~,+ > f i ~ V i = 1, 2 . . . . . .~.
(6) (7)
A global minimum can be found using Lagrange multipliers. Since ~-~. is continuous, inequality (6) can be replaced by an equality, and we will show that constraint (7) is redundant. Thus, the problem is minimizing (5) subject to (6). The closed form solution can be shown to be:
~it =fit + ~j=l d~jf~t "ytAmax
-
i = 1, 2 . . . . . .S".
J
fit ~ -- Ci,t_l, (12)
A. Dutta, Y.K. Kim / European Journal of OperationalResearch 91 (1996)395-410 Table 11 Unit costs of five expansion options for link length of 100 km Expansion option
capacity (Kbps)
Discrete unit cost a ($/Kbps)
Lower envelope ($/Kbps)
1 2 3 4 5
5 10 20 40 80
2000 1800 1700 1400 1000
1000 1000 1000 1000 1000
a Unit cost = (fixed cost + variable cost * 100 km)/capacity. Data of fixed and variable costs are given in Table 5. Since fit, di, 7t a n d d m a x have positive values, we have
gt~it + Ci,t-1 - f i t > O, satisfying c o n s t r a i n t (7). U s i n g Co0, the starting n e t w o r k capacity, expression (12) can b e used to c o m p u t e the e x p a n s i o n s of all successive periods.
4.3 Comparison between heuristic and benchmark solutions Note that the c o n t i n u o u s l i n e a r lower envelope is a relatively good a p p r o x i m a t i o n of the discrete capacity cost f u n c t i o n for large e x p a n s i o n o p t i o n s b u t n o t the smaller ones. T a b l e 11 illustrates this fact u s i n g cost data from T a b l e 5. T a b l e 12 a n d 13 c o m p a r e total costs o b t a i n e d by E X P - P L A N a n d the b e n c h m a r k u s i n g the 15-node p r o b l e m . T h e difference q u i t e low w h e n traffic growth rate is large o r Ama x is small. This is b e c a u s e in b o t h c i r c u m s t a n c e s large e x p a n s i o n Table 12 Comparison of heuristic and benchmark for different Amax Zlmax
Delay limit
PR c o s t (KS)
Benchmark cost ( K S )
Cost difference (%)
0.025 0.030 0.035 0.040 0.045 0.050
1779 1391 1175 1027 921 791
1741 1364 1096 894 738 612
2.2 2.0 7.2 14.9 24.8 29.2
Initial link capacity: 100 Kbps; Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 packet/sec.
409
Table 13 Comparison of heuristic and benchmark for different traffic growth rates Annual traffic growth rate (%)
PR cost (KS)
Benchmark (KS)
Cost difference
5 10 15 20 25 30
567 765 1027 1248 1584 1995
423 660 894 1170 1492 1854
34.0 15.9 14.9 6.7 6.2 7.6
Initial link capacity: 100 Kbps; Amax:0.04 sec. Average packet size: 1000 bits. Initial traffic between each node pair: 3.0 pkt/sec.
o p t i o n s are n e e d e d , for which the l i n e a r e n v e l o p e is a relatively good a p p r o x i m a t i o n . T h e cost diff e r e n c e is higher for large Amax or low traffic growth rates w h e n smaller sized e x p a n s i o n options are used. I n fact, T a b l e s 12 a n d 13 show differences a r o u n d 30%. W e believe that these i n s t a n c e s are c a u s e d by poor a p p r o x i m a t i o n of the lower e n v e l o p e r a t h e r t h a n poor p e r f o r m a n c e by E X P - P L A N . T a b l e 14 shows that the gap b e t w e e n E X P P L A N a n d b e n c h m a r k costs b e c o m e s smaller as the n u m b e r of n o d e s in the n e t w o r k increases. This suggests that the a p p r o x i m a t i o n of the heuristic may actually be i m p r o v i n g as the network gets larger. Actually, E X P - P L A N yielded slightly lower cost t h a n the b e n c h m a r k for o n e p r o b l e m i n s t a n c e ( u n d e r l i n e d in T a b l e 14). O n e Table 14 Comparison of heuristic and benchmark for varying network size Network size
PR c o s t (KS)
Benchmark cost ( K S )
Cost difference
5 15 30 50
124 1027 1507 3076
82 894 1357 3112
51.2 14.9 11.1 -0.1
Initial link capacity: 100 Kbps; Amax:0.04 sec. Traffic growth: 15%/yr. Average packet size: 1000 bits. Initial traffic between each node pair: 5 nodes: 25 pkt/sec. 15 nodes: 3.0 pkt/sec. 30 nodes: 0.7 pkt/sec. 50 nodes: 0.3 pkt/sec.
410
A. Dutta, Y.K. Kim / European Journal of Operational Research 91 (1996) 395-410
reason for this occurrence lies in i m p l e m e n t a t i o n o f the F D A . Even t h o u g h the F D A theoretically finds an optimal routing, its gradient n a t u r e makes c o n v e r g e n c e slow. F o r this reason most i m p l e m e n t a t i o n s stop searching w h e n i m p r o v e m e n t in average delay is less than a specified ' t o l e r a n c e ' [5]. T h e F D A is used by the b e n c h m a r k as well as E X P - P L A N , and the ' t o l e r a n c e ' m a y occasionally p r o d u c e such an u n e x p e c t e d result. H o w e v e r these occasions arose very infrequently during testing. W h e n they do, it is just an indication that the heuristic and b e n c h m a r k solutions are so close that the tolerance in F D A ' s termination condition can give rise to such anomalies.
5. Conclusion This p a p e r suggests a heuristic a p p r o a c h to multiperiod network capacity expansion, which consists o f first determining a myopic expansion plan based on single period heuristics, which is then subsequently rescheduled for e c o n o m i e s of scale effects. Several variations o f this basic idea can be implemented. F o r instance, an integer p r o g r a m m i n g formulation like [6] m a y be used to d e t e r m i n e the expansion plan of each period in isolation. Alternatively the F D A m a y be used in conjunction with cut-saturation heuristics [8] for the same purpose. Some caution must be exercised in using the b e n c h m a r k to judge solution quality of E X P - P L A N . As observed previously, the linear lower envelope will be a p o o r approxim a t i o n for those specific instances in which the cost structure is sharply ' b o w e d ' . T h e r e f o r e we suggest that in these situations, it may m o r e practical to c o m p a r e the heuristic solution against several b e n c h m a r k s obtained using continuous linear approximations that lie within a lower and u p p e r envelope o f the actual discrete cost function.
Acknowledgment T h e authors wish to t h a n k a n o n y m o u s refe r e e # 2 whose detailed c o m m e n t s were very helpful in revising the paper.
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