Technische Universität München
Lehrstuhl für Kommunikationsnetze
Prof. Dr.-Ing. Jörg Eberspächer
Bachelor’s Thesis
Demand Uncertainty in Multiperiod Network Planning
Author:
Thilo Schöndienst
Matriculation Number: 2838050
Address:
Christoph-Probst-Str. 12
80805 München
Germany
Email Address:
[email protected]
Supervisor:
Clara Meusburger
Begin:
01.09.2009
End:
11.12.2009
Abstract
In multiperiod planning for optical networks, future developments of demand, cost, and
other planning parameters are considered, in order to decide upon when and where to
implement and upgrade network equipment. Thus, with perfect knowledge of the future a
cost optimal solution can be achieved. Evidently though, the future is uncertain. In this
thesis an approach known as stochastic programming is used to increase robustness against
random fluctuations in demand and equipment cost. Several future scenarios are considered and weighted with probabilities. It is shown that by using stochastic programming
the robustness of multiperiod network planning concerning different types of uncertainty
(demand allocation, demand increase, development of equipment cost) can be increased
and infeasibilities in routing can be prevented.
iii
Kurzfassung
Bei der Planung optischer Netze über mehrere Perioden hinweg, werden die zukünftigen
Entwicklungen von Verkehrsaufkommen, Kostenentwicklung und weiterer Parameter
berücksichtigt, um zu entscheiden, wann, wo und welche Netzwerkkomponenten installiert oder erweitert werden. So kann bei genauer Kenntnis der Zukunft eine, in Hinblick
auf die Kosten, optimale Lösung berechnet werden. Die Zukunft jedoch ist ungewiss. In
dieser Arbeit wird die Stochastic-Programming-Methode verwendet, um die Robustheit der
Planung gegenüber Schwankungen in Verkehrsanforderungen und Komponentenkosten zu
erhöhen. Dazu werden mehrere Szenarien für die zukünftige Entwicklung erstellt und mit
Wahrscheinlichkeiten gewichtet. Es wird gezeigt, wie unter Einbeziehung der stochastischen Eigenschaften von Zukunftsvorhersagen, die Robustheit der Netzplanung gegenüber
verschiedenen unsicheren Faktoren (örtliche Verkehrsunsicherheit, Unsicherheit im Anstieg
des Verkehrsaufkommens, Unsicherheit bei den Kosten der Netzkomponenten) erhöht werden kann. Desweiteren können Blockaden in der Verkehrslenkung, ausgelöst durch hohen
Anstieg des Verkehrs, durch vorausschauende Planung vermieden werden.
v
Contents
1 Introduction
1
2 State of the Art
2.1 Multiperiod Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Decision making Under Demand Uncertainty . . . . . . . . . . . . . . . . .
2.2.1 Stochastic Programming . . . . . . . . . . . . . . . . . . . . . . . .
3
3
10
12
3 Modeling of Uncertainty in Multi Period Planning
3.1 Representation of Uncertain Development . . . . . . .
3.2 Stochastic Programming Model for Uncertain Demand
3.3 Stochastic Programming Model for Uncertain Cost . .
3.4 Structure of Results . . . . . . . . . . . . . . . . . . . .
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4 Results
4.1 Allocation Uncertainty . . . . . . .
4.2 Uncertainty in Demand Load . . .
4.3 Variations in Cost Decrease . . . .
4.4 Preventing Infeasibilities in Routing
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5 Conclusions & Outlook
39
A Abbreviations
41
List of Figures
43
List of Tables
45
Bibliography
47
Chapter 1
Introduction
Operators of optical networks aim for only two things: cost efficiency and functionality. To
provide both for all of the network’s lifetime it is necessary to plan ahead. Specifically, large
scale backbone networks come with a high amount of capital expenditures (Capex), aiming
to achieve future profits, and are focused at being operational for a long time. However,
parameters having an impact on planning can change significantly during operation time;
demand varies constantly, new technologies emerge, equipment prices may decrease or rise
due to certain economic circumstances. These changes are a challenge for network planners,
nonetheless to make the most of it variations should be taken into account by using multi
period planning. Moreover, the uncertain nature of these changes can be accounted for by
applying stochastics to the planning.
Minimizing costs while maximizing operability can be stated as an optimization problem
as follows: For every demand requirement between two nodes an appropriate route for the
lightpath from the source node to the target node has to be identified. Depending on this
routing the equipment (nodes, transponders, amplifiers, and regenerators) is placed in the
network [MSE08]. The goal is to find the most cost efficient feasible solution. In this thesis,
the demand always must be satisfied. Other modeling possibilities including addition of
penalty cost for blocking or capacity leasing [Leu05], [VCPD07] are not addressed. These
approaches would result in reducing the importance of feasible routing, focussing on cost
efficiency.
This thesis’ aim is to include Stochastic Programming into already implemented multi
period approaches. The motivation is to increase robustness of planning, that is to reduce
the negative effects of stochastic changes in demand and other parameters. Another aspect
of interest is what kinds of uncertainty there are, that is which parameters have random
characteristics. A differentiation is made, how severe the impact of different uncertain
factors is.
This thesis is organized as follows: Chapter 2 gives an introduction to the ILP formulation
of the network planning problem. In Section 2.1 the multiperiod approaches used for our
2
CHAPTER 1. INTRODUCTION
case studies are discussed. An introduction to planning and decision under uncertainty
is given in Section 2.2. The Stochastic Programming approach is introduced in Subsection 2.2.1. In Chapter 3 the multiperiod approaches are extended to include uncertainty.
Chapter 4 gives the findings of the conducted studies with Stochastic Programming models
and compares cost efficiency with others. Chapter 5 provides a conclusion and an outlook.
Chapter 2
State of the Art
2.1
Multiperiod Planning
To calculate optimal network solutions a method widely favored in literature is Integer
Linear Programming (ILP)[Leu05]. This term refers to an optimization problem with a
linear objective function subject to linear constraints.
Maximize cT x
Subject to Ax ≤ b.
x represents the vector of variables (to be determined), while c and b are vectors of (known)
coefficients and A is a (known) matrix of coefficients. The expression to be maximized or
minimized is called the objective function (cT x in this case). The equations Ax ≤ b are
the constraints which specify a convex polytope over which the objective function is to be
optimized.
The solution of the optimization has to consist of integer values, due to the fact that a
piece of network equipment can only be installed as a whole. Simply calculating results
allowing real numbers, then rounding up to the next integer would cause overcapacity.
Thus, no optimal solution would be achieved.
To calculate the optimal routing and equipment deployment strategy using ILP, it is necessary to express the problem in a mathematical way first. It is then transformed into a
program written in ‘A Math Programming Language’ (AMPL). AMPL is a suitable language because of its syntax being very close to a mathematical notation. In addition the
widely used CPLEX solver is used to do the actual optimization. It employs Branch and
Bound algorithms to solve ILP problems [IBM09].
The network is modeled as a undirected graph G = (V, E) where V is the set of nodes and
E is the set of edges. The required end-to-end demand matrices
DEM [np × np]
4
CHAPTER 2. STATE OF THE ART
are of granularity one, dnp indicates the demand between node pair np.
The set of node pairs with an entry in the demand matrix
N = {np1 , . . . , npn } := {np = {v, v 0 } : v, v 0 ∈ V, v 6= v 0 }
is designed.
A set of predefined paths
Pnp,p ⊂ E∀np ∈ N ∀p ∈ {1, . . . , noOfPaths np }
is defined where the parameter p gives the number of a certain path between the node pair
and is an element of the set {1, . . . , noOfPathsnp }. To maintain a reasonable simulation
time, this set of paths considered is a subset of paths containing only the ten shortest
possible ones for each nodepair.
An Optical Channel OChnp,p indicates the channel between nodes np alongside path number p. One OCh is needed to transport a demand of value one, specified in the demand
matrix. An Optical Multiplex Section OM S e denotes one installed optical multiplex section on edge e. This system is capable of connection nodes. It must be installed alongside
the path an OCh takes. One OM S can be used by multiple OChs depending on the
capacity of the equipment used.
The network model is now completely defined. To do an optimization however, an objective
function and conditions and must be defined. The goal of the optimization is to calculate
the lowest possible cost for the network, hence it is called the cost function. We assign
prices to crucial network elements, that are: nodes, transponders, amplifiers, regenerators,
and fiber.
Prices correspond to the NOBEL2 cost model [HGMS08]. They are normalized to the costs
of a 10Gbit/s transponder with a maximum transmission length of 750 km. The components available are bidirectional, capable of 80 wavelengths. An overview of equipment
costs is given in Table 2.1.
In our model each optical channel needs two transponders (with cost ct). If the length of
the channel exceeds the transparent transmission length, regenerators (with cost cr) are
installed. The length of a channel is the sum of the traversed edge lengths plus a penalty
for every transparent node passed through. Each installed OMS has fiber costs cf iber
containing costs for components: Optical Line Amplifier (OLA), dispersion compensating
fiber (DCF), and the dynamic gain equalizer (DGE).
DCF and Optical Line Amplifier are installed in the same interval on the fiber. A Dynamic
gain equalizer is needed at every fourth OLA site. The node costs cnode depend on the
required nodal degree. The nodal degree equals the number of OMS connected to a certain
node and cannot exceed the degree of 10. The number of installed nodes is limited to one
node per nodal site.
2.1. MULTIPERIOD PLANNING
5
Equipment
Abbr. in cost Reach
function
Optical
Line
Amplifier
Dispersion
compensation
fiber
Dynamic gain
equalizer
Nodal switch d
=1
Nodal switch d
=2
Nodal switch d
= 3, 4, 5
Nodal switch d
= 6, ..,10
10G transponder
10G regenerator
cf ibere
cf ibere
1500 km 3000
km
1500 km 3000
km
Cost
value
relative to 10G
transponder
with
750km
reach
2.77 3.45
0.728 0.88
cf ibere
3.17
cnoded
10.83
cnoded
25.30
cnoded
10.42 x d +2.75
cnoded
11.11 x d +2.75
ct10G
cr10G
1500 km 3000
km
1500 km 3000
km
1.25 1.67
1.75 2.34
Table 2.1: Relative cost values provided by NOBEL 2 multilayer cost model
In Figure 2.1 the components are put into context. Visible are two nodes, connected
by one OMS containing one OCh. Thus there are two transponders, one at each node.
Along the OMS there are four Optical Line Amplifiers each with an additional Dispersion
Compensation Fiber. On the fourth OLA site there is a Dynamic Gain Equalizer installed.
The presence of a regenerator means the channel length exceeding the transparent length.
Note that although the maximum nodal degree today is lower we assume technology to
evolve, enabling higher nodal degrees in future.
The capacity constraint 2.1 limits the number of OChs per edge to the maximum number
of wavelengths times the number of installed OMS.
X
OChnp,p ≤ noOfWavelengths × OM S e
∀e ∈ EDGES
(2.1)
np,pnp :e∈P
The demand transported constraint 2.2 ensures an OCh is present for every demand entry
6
CHAPTER 2. STATE OF THE ART
Figure 2.1: Illustration of components needed to connect two nodes with one Optical
Channel on one Optical Multiplex Section
in the demand matrix.
X
OChnp,p = dnp
∀np ∈ N
(2.2)
pnp
The objective function to minimize is the cost function Cost. It is composed of the sum
of the cost of each part of equipment needed for the solution imposed by the constraints.
Cost =
X
OM S e × cf ibere
e
+
X
OChnp,p × ctnp,p × 2
np,pnp
+
X
OChnp,p × crnp,p × noOf Regeneratornp,p
np,pnp
+
X
noded,v × cnoded
(2.3)
d,v
If an integer solution is found for the optimization problem, the values we are interested
in are OM Se , OChnp,p and the resulting nodal degree of each node. These results fully
define the placement of equipment needed to satisfy the demand.
The optimization is written in single period notation, the planning results in a network
satisfying one demand matrix DEM . If multi period planning is done an index t ∈
2.1. MULTIPERIOD PLANNING
7
PERIODS is added to time dependent variables. Typically one period is a fixed amount
of time, for example a year. It denotes the time that passes between upgrading of network
equipment to satisfy changing demand.
We limit the optimization to routing and skip wavelength assignment to save calculation
time. Complexity is greatly reduced by leaving out the assignment of colors to OChs. The
remaining routing problem still has rather long runtimes.
Depending on the chosen approach (see Table 2.2), all demand matrices are provided in
advance or sequentially one period of time after the other. Optimization software then
first calculates if the problem is feasible with the input data given or not. If the problem
is feasible a non–integer solution is calculated, giving a lower bound for the solution.
Iteratively integer–solutions are computed until optimality is achieved. Conveniently for
the solver a gap criterion, expressed as a percentage, may be defined. If a solution is found
within this percentage around optimality the iterations are stopped to save simulation
time.
The optimization process is done, according to the chosen approach, once for all periods
together (for approaches All Periods and Stochastic Programming), incrementally one period at a time (for the Incremental approach). In a combination of the two calculation is
done repeatedly for all periods together every time a change of the input data occurs (All
Periods with Deviating Demand approach). If the optimization is done incrementally and
infeasibility occurs a time after period one, blocking occurs, demands can not be routed
and planning has failed.
In the model developed, one time period can equal any chosen amount of time, for example
year. Shorter durations may increase flexibility and a planning at mid-year, or even shorter
may be established.
During the planning horizon, as mentioned earlier, parameters vary. Observations have
shown that equipment prices are generally gradually reduced, due to new technologies
emerging. Therefore we use a declining-cost model with start values from [HGMS08] and
a cost decreasing factor (cdf ). Depending on approach and studied effect of uncertainty
on results the cdf may remain constant or vary with time (see Section 3.3).
Apart from the nodal degree development future emerging technologies are not included
in this thesis.
Table 2.2 shows some characteristics of different approaches used in this study. The
ILP code for Incremental and All Periods approaches has been developed in connection
with [MSE08]. Using the Incremental, the cost function is optimized for one period after
another. The expenses per period are hence the lowest possible regarded individually. The
future is not concerned in planning but only the current periods parameters: the Incremental approach is no true multi period approach. The All Periods approach optimizes
the cost function for all periods, for which forecasts are available at once, leading to an
optimal overall result. When simulation is done with the All Periods Deviating Demand
8
CHAPTER 2. STATE OF THE ART
Incremental:
Only the demand known, i.e., the recent period’s demand, is used for
the network planning. Running an optimizer on the cost function, cost
optimality is achieved for a single time period. However the fact that
previously installed channels remain fixed in place, leads to an overall
solution (concerning more than one period of time) that is typically
suboptimal.
All Periods planning:
The All Periods approach minimizes network costs over multiple periods of time for given forecasts. Demand matrices are estimated in
advance, one for each time step, e.g., a year. Parameters of the cost
function are assigned additional indices t for time. The optimization
is done for all t ∈ noOf P eriods at once; hence the solution is optimal
for the time considered, assuming the forecast is correct.
All Periods planning with deviating development:
The All Periods model gives an optimal solution. In a real world
setting however the actual demand is unlikely to be known a couple of
time steps in advance. Accepting this lack of knowledge the approach
is revised; after each time step, the demand estimated in advance is
replaced with the realized actual demand. For the corrected values an
optimal solution for the rest of the lifetime is calculated. This revision
step is done after each period. Additionally the remaining forecasts
may be replaced with newer ones, to achieve better solutions.
Stochastic Programming: (see Chapter 3)
The uncertainty of future development is taken into account by assuming not one but a set of possible evolutions is known. A solution
for the deployment and routing is calculated that minimizes the total
expected costs given the probabilities of the alternative developments
(called scenarios)
Table 2.2: Different Multiperiod Approaches Used in This Thesis
2.1. MULTIPERIOD PLANNING
9
model, first an All Periods simulation is calculated. For the following periods, the solutions
of the previous planning are fixed. Then the input Values for Demand are changed and
based upon them a new All Periods solution is calculated. This procedure is repeated every
planning period. The contribution of this thesis is the Stochastic Programming approach.
The decision which approach is used for a special planning problem depends on the planning
conditions and aims, see Chapter 4 for an attempted differentiation.
Different authors propose a variety of consideration on this major planning topic. Reference
[SKS06] and [MSE08] discuss a variety of multi period approach, however they leave the
inclusion of uncertainty to further research. [AKP03] discusses uncertainty in capacity
expansion in general but does not include with the special properties of network routing.
[AC06] propose algorithms to achieve robust and nearly optimal routing with minimal to no
knowledge of demands. Cost efficiency however is not concerned. [LZS05] minimizes hop
count with only partial knowledge of demand, using load balancing. However multi period
and budget considerations are not made. [KLO+ 03] uses stochastic programming to tackle
routing under uncertainty for a given budget. Long term planning however is not addressed.
[AZ07] extends single-stage robust optimization to two stages. Multi period considerations
are not made. [Sch06] assumes demand to be entirely uncertain, no forecasts possible.
At the same time it is stated that a network operator should exploit all the information
on trends of future traffic behavior and plan the network accordingly. [MW05] proposes
a stochastic approach to maximize revenue under uncertain demand conditions. Capex
minimization for multi period investments is no concern, though. [ZMLB08] studies the
impact of uncertainty in physical parameters on dimensioning of optical networks.
In the proposed Stochastic Programming Multi Period approach cost efficient multi period
planning for uncertain developments is done.
10
CHAPTER 2. STATE OF THE ART
2.2
Decision making Under Demand Uncertainty
The inclusion of uncertainty in network planning adds to the problem of deciding upon the
exact layout of how to build or upgrade a network.
According to [BHS99] a good decision is based on logic, it considers all available data and
possible alternatives, and applies a quantitative approach. There is no guarantee, however,
a good decision will result in a favorable outcome. A bad decision is one that is not based
on logic. Sometimes a bad decision will provide good results, but it is still a bad decision.
Although occasionally good decisions yield bad results, using decision theory will result in
successful outcomes in the long run.
“For every decision, the steps that need to be taken in order to make it a good decision
are basically the same:
1. Clearly define the problem at hand.
2. List the possible alternatives.
3. Identify the possible outcomes.
4. List the payoff or profit of each combination of alternatives and outcomes.
5. Select a mathematical decision theory model.
6. Apply the model and make your decision.” [BHS99]
To decide upon which model to apply, the environment of the decision must be defined.
This involves taking into consideration both the amount of risk and uncertainty involved.
There are three major categories of environments.
• The first is Decision Under Certainty, there every consequence of possible decisions
is known for sure. Every realization of a state of the world sj is fully defined. The
alternative, or the combination of decisions can easily be chosen to maximize profit.
• Decision Under Risk is equal to having probabilities pj for realizations of states sj .
Here randomness is included in the chain of cause and effect.
• The most severe case will be called Decision Under Ambiguity: Possibly realizing
states sj are known, their probabilities however are not.
For decision under risk, expected values of consequences of all decisions can be calculated
easily since all probabilities are given. Decision under ambiguity can be addressed with
the principle of indifference. The principle states, that if no knowledge is available about
the probabilities of outcomes, and there is no knowledge indicating unequal probabilities
(John Maynard Keynes), every possibility is assigned the same probability of N1 : (N =
number of possible outcomes). With this assumption made, the expected value can be
2.2. DECISION MAKING UNDER DEMAND UNCERTAINTY
Level I:
A ClearEnough
Future
Level II:
Alternative
Futures
Level III:
A Range of
Futures
Level IV:
True
Ambiguity
11
A single forecast is suffices for network planning. Uncertainty is neglectable, cost and effort to include it would
outsize the possible gain. This was valid for example in
past telephone networks.
A handful of alternatives can be identified. These discrete scenarios are assigned probabilities, possibly not
easy to quantify.
Here no distinguishable discrete scenarios can be given.
The alternatives are countless, and a large number of
scenarios merely define some boundaries.
No reasonable prediction can be made in this case. Possibly not even the dimension of uncertainty is obvious.
Table 2.3: Levels of Uncertainty According to Grover [LG05]
calculated. The later on introduced Stochastic Programming approach uses the expected
value for optimization. It can therefore be applied in both cases.
A slightly different approach by [LG05] divides uncertainty in four levels(Table 2.3). Level
I clearly belonging to decision under certainty and Level II meaning a decision under
risk. Uncertainty of Level III still allows a decision to be made applying the principle of
indifference. Level IV however implies that the decision can not be made on assumptions
about the future but must be made independently. An approach to do network planning
under these conditions is presented in [Sch06].
Once the environment has been identified, the appropriate mathematical decision model
has to be chosen. Suitable models for Decision Making Under Risk are, for example EMV
(expected monetary value), EVPI (expected value of perfect information), EOL (expected
opportunity loss), and Sensitivity Analysis. All these are based on some kind of expected
value calculation. If the environment proves to be ambiguous appropriate models include
Maximax, Maximin, and Minimax.
As economical decision theory lies beyond the scope of this thesis, further investigations
on mathematical decision models are not made.
In the global market economy a variety of additional methods can be used to simulate
a network construction project. If other forces influencing parameters become involved,
the theoretical field changes from stochastics towards game theory. Here, other players
are included and simulation becomes much more complex. Influences can come from competitors, contractors, consumers, and customers. In [VCPD07] an approach utilizing Real
Options is explained in detail. This approach involves someone willing to grant the options
and a revenue from sold or let capacity. Regarding flexibility a high gain is possible from
having options on capacity leases for extreme scenarios. Achieving robustness against fluc-
12
CHAPTER 2. STATE OF THE ART
tuations in demands is possible without the otherwise necessary wasting of capacity trough
overprovisioning. The Stochastic Programming approach developed in this thesis may well
be used to calculate the price of an option, or to give the value of an insurance, covering
for penalty expenses.
2.2.1
Stochastic Programming
Deterministic optimization models do not represent reality. However a decision making
tool is of little value if the underlying simulation is unrealistic. One well known way of
dealing with optimization problems under uncertainty is Stochastic Programming [KW94].
Multiple disciplines take advantage of this approach. Stochastic Programming for example
benefits financial applications like portfolio optimization or applications from operational
research like fleet management or inventory problems [van07].
The interpretations of Stochastic Programming are manifold. Generally speaking it means
the inclusion of stochastic parameters in a linear program. Hence, Stochastic Programming
provides means to include alternative future outlooks and to weight each with a probability
of occurrence, thus it approximates reality much better than deterministic approaches.
Linear programs are problems that can be expressed in canonical form:
Maximize cT x
Subject to Ax ≤ b.
x represents the vector of variables (to be determined), while c and b are vectors of (known)
coefficients and A is a (known) matrix of coefficients. The expression to be maximized or
minimized is called the objective function (cT x in this case). The equations Ax ≤ b are
the constraints which specify a convex polytope over which the objective function is to be
optimized.
The conversion to a stochastic program can be done by simply adding a random vector ξ
to the constraint: Ax ≤ b + ξ.
Consequently the goal of the optimization changes to the expected value E(cT x)
Thus the Stochastic Program is:
Maximize E(cT x)
Subject to Ax ≤ b + ξ.
Chapter 3
Modeling of Uncertainty in Multi
Period Planning
The goal of multiperiod planning is to deal with time dependent planning parameters
in a cost efficient way. Throughout the operation time of a network several parameters
which are influencing the optimal planning solution are changing. A single period planning
approach can not address these changes which are occurring at some point in network
lifetime. More efficient solutions can be examined by means of multiperiod planning. In
previous studies the development of parameters over time has been assumed to be known for
sure. These have shown interesting insights into network planning with cost minimization
in mind [MSE08]. As mentioned in [MSE08] [SKS06] further challenges of multiperiod
approaches lie in the parameters’ unpredictable nature. While the current situation is
known to a high degree, statements about future developments can never be made with
certainty. The contribution of this thesis is to include the uncertainty of parameters into
multi period planning models.
3.1
Representation of Uncertain Development
To be able to deal with parameter development in a mathematical way a representation is
needed that can be incorporated into the simulation models.
Since there is no way of calculating a solution, optimal in any way, with true ambiguity
(see Table 2.3), facilitating assumptions have to be made. The severity of the uncertainty
has to be reduced to a level where alternatives can be specified and their respective probabilities be estimated. These different development forecasts are called scenarios. Where
different scenarios are concerned a widely used representation is a tree shape as pictured
in 3.1 called Scenario Tree [AKP03] . The widening tree structure with a number of nodes
increasing with time represents the assumption that reality is known fairly accurate in the
14 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING
31.1
Time
1
21
31.2
1
32.1
22
Planning for
Period 1
Planning for
Period 2
Planning for
Period 3
32.2
Figure 3.1: Sample scenario tree spanning three periods depicting four scenarios; scenario
1 is highlighted
near future and less further away. Also a dependency of developments on previous ones is
considered a realistic model [DCW00].
For the scenario trees used in this thesis we assume the structure given in Figure 3.1. Each
node represents a possible state. In the case of demand uncertainty this means for each node
N (p) with p ≥ 1 a demand matrix DEM (p, s) with (p ∈ P ERIODS, s ∈ SCEN ARIOS)
is given. If uncertainty in the decrease of equipment cost is regarded, for each node a cost
decreasing factor cdf (p, s) with (p ∈ P ERIODS, s ∈ SCEN ARIOS) is needed. Planning
is done at time zero. The first period contains only one node since the initial state of
the network is known. Future periods contain 2(p−1) alternative nodes. This values are
chosen to create scenarios that can easily be differentiated and to reduce simulation time by
reducing complexity. For a closer approximation more nodes, and more scenarios may be
included, this does however not change the observations made in this thesis. It leads to a
number of possible outcomes that increases with time. Therefore we act on the assumption
of a more uncertain development the further a prediction goes.
Leaf-nodes, i.e., nodes without emerging edges, can be reached by only one single possible
path from the root, i.e., the starting point to the very left. This path forms a scenario.
In Figure 3.1 an example scenario is highlighted in red. It is called Scenario 1, as scenarios
are always numbered from top of the tree to the bottom in this thesis. Three periods
are given with four leaf nodes, that is four alternatives in the final stage. Therefore four
scenarios are resulting from this tree.
In the following uncertainties in two of the parameters which have an influence on the
3.2. STOCHASTIC PROGRAMMING MODEL FOR UNCERTAIN DEMAND
15
optimal solution of the planning problem are concerned. One being uncertainty in the
development of demand, the other being uncertainty in the development of equipment cost
modeled by the common cost decreasing factor.
3.2
Stochastic Programming Model for Uncertain Demand
A popular interpretation of stochastic programming for multi period simulations is a two
stage model, meaning two periods are covered by the approach. Here an immediate problem
is solved, with knowledge about possible futures in mind. For the possible futures already
appropriate reactions are given. Depending on the actual realization of the future, these
given solutions have to be applied. We adapt this idea and extend it to a three stages, or
three periods.
However the special property is that by means of nonaticipativity constraints a single solution is found for the first stage. This single solution prepares the path well for any
possible future, optimizing the expected value of the outcome. A commonly used representation of these alternative developments are scenario trees [DCW00] see Figure 3.1 for
an illustration. This way the result
In the course of this thesis additions and alterations to the original model from [MSE08]
are made. This model is explained in Section 2.1. The changes are made to the All Periods
variant of the model, that optimizes the cost function for all considered time periods at
once. This way all the time dependent parameters and variables were already in possession
of an index for time. The modifications were made roughly according to [FGK03, ex 4.5]
in the following order:
1. To include multiple scenarios, the demand matrices’ dimension is extended by one
to cover multiple scenarios. The variable s denotes a scenario)
DEM (t, s) = [dnp ]np×np t ∈ P ERIODS, s ∈ SCEN ARIOS
(3.1)
The input data for the demand is extended by one dimension representing the scenarios. A demand matrix is created for each scenario in each period. However due
to the nature of the chosen scenario tree, in the first period demand matrices for all
scenarios are equal. In the following stages the number of unique matrices equals to
the number of nodes in the tree at that stage. (e.g. in a model of 3 periods with four
alternative outcomes in stage 3 at stage two there are two unique demand forecasts)
2. A parameter that contains the probabilities of the individual scenarios is added
prob(s) with (s ∈ SCEN ARIOS)
(3.2)
16 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING
3. The variables’ (node, OMS, OCh) dimensionis extended for them to result in solutions
for all scenarios
OM S(p, s, edge)
OCh(p, s, np, subset paths)
node(p, s, d, n)
≥ 0 integer
≥ 0 integer
binary
(3.3)
(3.4)
(3.5)
4. Adapt existing constraints to be imposed on each scenario
Capacity constraint:
X
OChnp,p,t,s ≤ noOfWavelengths × OM S e (t, s)
np,pnp :e∈P
∀e ∈ EDGES∀t ∈ P ERIODS∀sinSCEN ARIOS
Demand transported constraint:
subset paths(np)
X
(OCh(p, t, s, np, pathN o)) ≥ dem(t, s, np)
pathN o
∀t ∈ P ERIODS, s ∈ SCEN ARIOS, np ∈ DEM AN DP AIRS, p ∈ P AT HS
5. Add nonanticipativity constraints
3.2. STOCHASTIC PROGRAMMING MODEL FOR UNCERTAIN DEMAND
17
OM S(1, s, edge) = OM S(1, s + 1, edge)
(∀edge ∈ EDGES, s ∈ SCEN ARIOS)
∀edge ∈ EDGES : OM S(2, 1, edge) = OM S(2, 2, edge)
∀edge ∈ EDGES : OM S(2, 3, edge) = OM S(2, 4, edge)
OCh(1, s, np, pathN o) = OCh(1, s + 1, np, pathN o)
(∀s ∈ (SCEN ARIOS − 1), np ∈ DEM AN DP AIRS, pathN o ∈ subsetp aths(np)) :
OCh(2, 1, np, pathN o) = OCh(2, 2, np, pathN o)
(∀np ∈ DEM AN DP AIRS, pathN o ∈ subsetp aths(np)) :
OCh(2, 3, np, pathN o) = OCh(2, 4, np, pathN o)
(∀np ∈ DEM AN DP AIRS, pathN o ∈ subsetp aths(np)) :
node(1, s, d, n) = node(1, s + 1, d, n)
(∀s ∈ (SCEN ARIOS − 1), n ∈ N ODES, d ∈ DEGREE)
node(2, 1, d, n) = node(2, 2, d, n)
(∀n ∈ N ODES, d ∈ DEGREE)
node(2, 3, d, n) = node(2, 4, d, n)
(∀n ∈ N ODES, d ∈ DEGREE) (3.6)
The above are called nonanticipativity constraints. These make sure the result of the
optimization is not four different strategies but a single one for the next step and
resembling the scenario tree thereafter.
If more than tree periods with four scenarios having the given tree-structure are
considered these constraints have to be changed accordingly. The Constraints for
period two are only applicable to the chosen tree structure with two possible states
in period two. If these were not imposed, two decisions for each state might be given,
violating the objective to have only one decision based on the realized demand.
A real network operator is likely to repeat the network planning at each new period.
This is to take advantage of newly available forecasts and demand estimates. In this
case the nonanticipativity constraint for the first period is crucial, as one decision is
needed for the current period’s investment while the following can be neglected.
6. Change optimization goal to expected cost
18 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING
min E(costs) :
NS
X
p(s) ∗
s
+
p
DEM AN
DP AIRS subsetpaths(np)
X
X
np
+
OCh(p, s, np, pathN o) ∗ 2 ∗ costtrans (p, np, pathN o)
pathN o
N ODES
X DEGREE
X
n
OM S(p, s, e) ∗ costf iber (p, e)
e
DEM AN
DP AIRS subsetpaths(np)
X
X
np
+
NP EDGES
X
X
OCh(p, s, np, pathN o) ∗ costregen (p, np, pathN o)
pathN o
node(p, s, d, n) ∗ costnode (p, d) − node(p − 1, s, d, n) ∗ costnode (p, d)
d
(3.7)
as objective for the optimizer previously the cost of the components was used.
This was changed to the expected value meaning feasible solutions for the scenarios
weighted with their probabilities
3.3
Stochastic Programming Model for Uncertain
Cost
The above steps were taken but instead of indexing the parameters dependent on demand
over scenarios anything related to cost was indexed. Concerning the marked (†) steps the
existing models for Incremental Planning and All Periods had to be adapted as well.
1. †The constant cost decrease factor was turned into a matrix of cost development:
cdf [p ∈ P ERIODS × s ∈ SCEN ARIOS]
(3.8)
2. Add a parameter that contains the probabilities of the individual scenarios
3. Extend the variables’ (node, OMS, OCh) dimension to give solutions for all scenarios
4. Adapt existing constraints to be imposed on each scenario
5. Add nonanticipativity constraints
6. †The monotonically declining equipment cost parameters had to be adapted to the
circumstance that costs may develop differently.
3.4. STRUCTURE OF RESULTS
19
costnode (p ∈ P ERIODS, s ∈ SCEN ARIOS, d ∈ DEGREE)
costtrans (p ∈ P ERIODS, s ∈ SCEN ARIOS, np ∈ DEM AN DP AIRS, pathN o ∈ subset paths(n
costregen (p ∈ P ERIODS, s ∈ SCEN ARIOS, np ∈ DEM AN DP AIRS, pathN o ∈ subset paths(
costf iber (p ∈ P ERIODS, s ∈ SCEN ARIOS, edge ∈ EDGES) (3.9)
7. Change objective function to be the expected value of the total cost
min E(costs) :
SCEN
ARIOS
X
p(s) ∗
P ERIODS
X EDGES
X
s
+
p
DEM AN
DP AIRS subsetpaths(np)
X
X
np
+
OCh(p, s, np, pathN o) ∗ 2 ∗ costtrans (p, s, np, pathN o)
pathN o
DEM AN
DP AIRS subsetpaths(np)
X
X
np
N ODES
X DEGREE
X
+
n
OM S(p, s, e) ∗ costf iber (p, s, e)
e
OCh(p, s, np, pathN o) ∗ costregen (p, s, np, pathN o)
pathN o
node(p, s, d, n)∗costnode (p, s, d)−node(p−1, s, d, n)∗costnode (p, s, d)
d
(3.10)
Note the indices at the cost parameters.
For the inclusion of various cost scenarios in the Incremental approach only minor
modifications were necessary, since equipment costs were already calculated periodically inside a loop. A parameter which cdf to use in which period was all that had
to be added.
3.4
Structure of Results
has the shape of a tree, as shown in Figure 3.2.
At each period the decision has to be made according to what demand forecast has realized.
If new forecasts are available these can be considered for a new run of the Stochastic
Program. In this thesis however only initial greenfield planning is considered, giving the
decisions for the first three periods, based on four initially forecasted demand scenarios.
20 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING
OMS3.1.1 ; OCh3.1.1 ; node3.1.1
OMS2.1 ; OCh2.1 ; node2.1
OMS3.1.2 ; OCh3.1.2 ; node3.1.2
OMS1; OCh1; node1
OMS3.2.1 ; OCh3.2.1 ; node3.2.1
OMS2.2 ; OCh2.2 ; node2.2
OMS3.2.2 ; OCh3.2.2 ; node3.2.2
Figure 3.2: Illustration of results given by Stochastic Programming model
To compare various approaches in terms of total cost the result of the optimization can be
used without modification, since it already is the expected value. For previously existing
approaches, i.e., Incremental and AllPPeriods, it is necessary to calculate the costs scenario(p(s) ∗ costs(s)) to calculate the expected
wise first and then use E(costs) = SCENARIOS
s
value.
Chapter 4
Results
To compare the Stochastic Programming approach to the previously existing, several case
studies have been conducted. Additionally the influence of various kinds of uncertainty
on variations in total cost are of interest. It is shown that depending on the factor of
uncertainty the potential cost savings from multi period planning may vary. To study
the impact of various parameters of uncertainty, each of them has been isolated to allow
individual studies. Those factors are Uncertainty in Allocation of Demand, Uncertainty
in Increase of Demand and Uncertainty in Cost Development. Finally the network was
assumed to be highly loaded and it was tested whether the approaches led to results at
all. These sensitivity analysis offer a rough estimation of the performance of the stochastic
programming approach.
In Figure 4.1 an overview of the results is shown. The heights of the bars indicate the mean
expected cost of an approach relative to the optimum. For the expected costs calculation
equal likelihood of any scenarios to realize is assumed. The data used is the mean expected
value of all studies conducted for the particular uncertain parameter. The categories are
the afore mentioned three factors of uncertainty, explained in the following sections of this
chapter. Measures for Incremental and Stochastic Programming approach are the expected
costs calculated as explained in Chapters 2 and 3, respectively. All-Periods alway results
in one as it is equal to perfect knowledge of the scenario realizing and therefore gives the
cost optimal planning solution. This can never be realistically achieved since there is no
perfect knowledge of the future. It does however give the lower bound of costs. To the AllPeriods with Deviating Demand Approach two different measures are applied: Realistic
All-Periods considers planning for every of the four scenarios with realization of every of
the four scenarios. Using the All Periods with deviating demand approach (Table 2.2), this
results in sixteen possible cost results. In the chart, the expected value is given, assuming
that all combinations of forecasted/realized scenario have the same likelihood. Worst Case
All-Periods considers again all combinations of scenarios predicted/realized. Thus, for the
resulting costs of each realized scenario only the forecast most differing in total cost is
considered, that is the worst case is assumed to happen. The mean considers the four
22
CHAPTER 4. RESULTS
Expected cost for various uncertain parameters
Expected cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
Demand allocation
Demand increase
Cost decrease
Uncertain parameter
All Periods
Stochastic Programming
Realistic - All Periods
All Periods - Worst Case
Incremental
Figure 4.1: Overview of Cost-Dependency on Factor of Uncertainty
worst case results to be of equal probability.
In the general overview in Figure 4.1 the relatively high cost for Incremental planning is
obvious. All other approaches have expected costs closer to the optimum. Thus planning
ahead generally promises to be rewarding in terms of total expected cost. For Uncertainty in Demand Allocation and for Uncertainty in Cost Decrease the true multi period
approaches perform all very well, within one percent from the optimum. Considering
Uncertainty in Demand Increase the expected costs for these approaches are higher. Incremental planning does, however perform better than in the other cases of uncertainty
considered. This effect is due to overprovisioning, meaning excess capacity is provided. In
detail this is explained in Section 4.2.
The data used modeling the network is the USA-NSF-Net taken from [OPTW07] pictured
in Figure 4.2. Input demand values were modeled according to the scope of the particular
study. A steady increase in demand was assumed.
Palo Alto
CA1
WA Seattle
San Diego
CA2
Salt Lake City
UT
Houston
TX
Lincoln
NE
IL
Figure 4.2: US Topology used in models
Boulder
CO
Atlanta
GA
MI
PA
DC
NY
IL: Urbana Champaign
MI: Ann Arbor
PA: Pittsburgh
DC: Washington
NY: Ithaca
NJ: Princeton
NJ
~ 1000 km
23
24
4.1
CHAPTER 4. RESULTS
Allocation Uncertainty
In the first case studied, the allocation of the demand is assumed to be uncertain. The
parameters cost decreasing factor and the increase of the demand are assumed to be known,
thus the results are influenced only by the varying allocation.
It is assumed that half of the allocations of upcoming demands are known the other half
is randomly distributed. This is done to achieve comparable results. If the allocation was
completely unknown results would differ extremely for each run, randomizing the complete
distribution.
Period 1
Period 2
Period 3
DEM(3,1)
DEM(2,{1,2})
+
DEM(1,{1..4})
DEM(2,{1..4})
+
DEM(2,{3,4}
DEM(3,2)
+
+
DEM(3,{1..4})
+
+
DEM(3,3)
DEM(3,4)
Figure 4.3: Demand tree for allocation uncertainty
The demand tree is composed as in Figure 4.3. Only the additional demand per period is
shown here, the demand of the previous periods has to be added to get the total demand.
First year development is, as it is in any other simulation, assumed to be known. In the
second period the common allocation of a base demand (yellowish) is known. This base
demand is the same for both possible developments. Equally for period three demands one
common base demand for all four possibilities is assumed. On top of the base demands
an individual demand matrix, that about equals the base demand in terms of volume, not
allocation though, is added at each node of the scenario tree. All demand matrices are
4.1. ALLOCATION UNCERTAINTY
25
generated randomly and therefore it is possible, that the random demand adds demand to
a node already covered by base demand.
The values of demand are chosen in a way that no blocking occurs for any chosen approach
and a reasonable maximum nodal degree is reached. Which in our example is equal to an
initial demand of uniformly distributed values from zero to eight. In the following periods
the added demand is the sum of two uniform random distributions, the common base
demand and the additional individual demand. In period two both distributions assume
values from zero to two resulting in an overall distribution between zero and four, in the
third and last period they both range from zero to one. Summed up these values result in
the total demand, hence in the third period demands of up to 8 + 4 + 2 = 14 may occur.
Expected cost for uncertainty in demand allocation. Example 1
Expected cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
All Periods
Stochastic
Programming
Realistic - All Periods
Worst Case - All
Periods
Incremenal
Approach
Figure 4.4: Expected costs for uncertain demand allocation. Example 1
For the first case study results are shown in Figures 4.4 and 4.5. In Figure 4.4 the height of
bars equals the expected value of the results of the approaches used relative to the expected
value for All Periods, assuming equal likelihood of scenarios to realize. In Figure 4.5 the
cost for the individual scenarios is compared. The cost is expressed relative to the cost for
the optimal All Periods approach in all charts.
In Figure 4.4 it is obvious that the Stochastic Programming and the Worst Case AllPeriods approaches result in total cost close to the optimum for any of the four scenarios.
the deviation from the optimum reveals to be within one percent for the multiperiod
approaches. The Incremental approach results in expected costs roughly 10% above the
optimum.
26
CHAPTER 4. RESULTS
Detailed cost for uncertainty in demand allocation. Example 1
Cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
1
2
3
4
Scenario
All Periods
Stochastic Programming
Worst Case - All Periods
Incremental
Figure 4.5: Costs For Different Scenarios Of Uncertain Demand Allocation. Example 1
A close look at Figure 4.5 shows that the third Stochastic Programming bar, representing
the third scenario, is elevated above those for Stochastic Programming in any other scenario. While outperformed by Stochastic Programming in scenarios one, two, and three
Worst Case All-Periods is (though the differences are small) better in three. This result
is due to the fact that on further examination the randomly generated matrices for third
period demand in scenarios 1, 2, and 4 turned out to be highly correlated and quite different in 3. The special characteristic of Stochastic Programming is to consider probabilities.
This gives an explanation. Scenarios are assumed to have a probability pi = 0.25 each,
resulting in 75% combined for the similar scenarios 1,2, and 4. The distribution of the
network components is calculated in favor of these as they yield a higher combined probability. In case it turns out scenario three equals the actual development, third period
installments have to make up for suboptimal preparations in period 1 and 2. This might
be the considered trade off for the robustness achieved.
For a second example the calculation is done again with different, but still uniform, randomly distributed values. The results thereof are shown in figure 4.6 again the bars show
that Stochastic Programming and Worst Case All Periods behave inversely. Here scenarios
one, two, or three, four, can be considered together. The first group being slightly more expensive if Stochastic Programming is applied, the latter giving higher cost for Worst Case
All Periods. The grouping effect is due to the difference in period two demand matrices.
It is caused by the nonanticipativity constraints forcing the SP approach to result in only
4.1. ALLOCATION UNCERTAINTY
27
Detailed cost for uncertainty in demand allocation. Example 2
Cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
1
2
3
4
Scenario
All Periods
Stochastic Programming
Worst Case - All Periods
Incremental
Figure 4.6: Costs For Different Scenarios Of Uncertain Demand Allocation. Example 2
two possible period two decisions, a common for scenario one and two and a common one
for three and four.
28
4.2
CHAPTER 4. RESULTS
Uncertainty in Demand Load
DEM(3,1)=0,3*DEM(2,{1,2})
DEM(3,1)=0,4*DEM(2,{1,2})
DEM(2,{1,2})=0,3*DEM(1)
DEM(2,{1,2})=0,4*DEM(1)
DEM(3,2)=0,7*DEM(2,{1,2})
DEM(3,2)=0,6*DEM(2,{1,2})
DEM(1)
DEM(1)
DEM(3,3)=0,3*DEM(2,{3,4})
DEM(3,3)=0,4*DEM(2,{3,4})
DEM(2,{3,4})=0,7*DEM(1)
DEM(2,{3,4})=0,6*DEM(1)
Example 1
Example 2
DEM(3,4)=0,7*DEM(2,{3,4})
DEM(3,4)=0,6*DEM(2,{3,4})
Figure 4.7: Scenarios for uncertainty in demand increase
Another uncertain parameter considered is the development of the demand load. This is
a realistic case, as observations have shown the number of demands may rise by hardly
predictable rates. If with a simple All Periods model the demand is not predicted correctly
potential savings from smart dimensioning the network in early stages are wasted. Using the Stochastic Programming planning method, several scenarios of periodical demand
increase are included in planning.
In Figure 4.7 the demand scenario tree is shown. For the first PEriod, the demand is a
known uniform random distribution. The following periods the demand allocation is the
same. The demand matrix is multiplied by a factor ≤ 1 the following period and added to
the existing demand. To provide comparability the ‘mean’ at each node is a factor of 0.5.
This matches the values chosen in the allocation uncertainty experiment in the previous
section (Section 4.1). The word ‘mean’ in this context means the alternatives are 0.3 and
= 0.5) in the first example and 0.4 and 0.6 in the second one ( 0.4+0.6
= 0.5).
0.7 ( 0.3+0.7
2
2
Leading to a more (in the first example) or less (second example) diverse development.
The initial matrix is a random distribution with values zero to eight uniformly distributed
over the network . This provides feasibility for all approaches and a reasonable max nodal
degree at the end of the planning horizon.
A first glance at Figure 4.8 and Figure 4.9 reveals that the possible profit from planning is
smaller in this case, assuming known allocation and uncertain increase, than in the case of
partially unknown allocation. The performance of incremental is better with an expected
value of total cost only seven percent above the optimal All Periods cost’s expected value.
While for the allocation uncertainty the cost for SP, real AP, and WC AP were within
a one percent margin, this time SP is expected to cost 4% more than AP. Still a gain is
possible from planning ahead, since Incremental lags behind all of them.
4.2. UNCERTAINTY IN DEMAND LOAD
29
Expected cost for uncertainty in demand increase. Example 1
Expected cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
All Periods
Stochastic
Programming
Realistic - All Periods
Worst Case - All
Periods
Incremenal
Approach
Figure 4.8: Expected costs for Uncertain Demand Increase At Known Locations. Example
1
Expected cost for uncertainty in demand increase. Example 2
Expected cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
All Periods
Stochastic
Programming
Realistic - All Periods
Worst Case - All
Periods
Incremenal
Approach
Figure 4.9: Expected costs for Uncertain Demand Increase At Known Locations. Example
2
30
CHAPTER 4. RESULTS
Comparing the Stochastic Programming and Worst Case - AP/ Real - AP approaches
the high cost of SP in Figure 4.8 is obvious. This is due to the high variance between
scenarios. As pictured in Figure 4.10 the range of absolute total costs is quite large.
However probabilities for scenarios are chosen to be equal. A lot of overprovisioning takes
place in case for WC-AP as well as SP comparing the plan for scenario 4 with a realization
of scenario 1.
It is important to recall the definitions of Real - AP and Worst Case - AP given at the
beginning of this chapter. Real - AP seems fairly better than the Worst Case - AP in
Figure 4.8. Still then equal probabilities for all scenarios and equal probabilities to plan
for the right scenario are rather unrealistic. The area between the two values gives an
estimate of a realistically achievable solution.
If the differences between scenarios are not as big the performance of SP significantly
improves and costs are expected to be below AP planning with a possibly wrong assumption
of future development. This is the case in the uncertain increase study, pictured in 4.9.
Characteristics of the SP approach can be derived from Figure 4.11.The scenarios with the
higher increase towards the end of forecast horizon, namely scenarios two and four, result
in lower cost. Scenarios three and four are less expensive than one and two, respectively.
The effect is caused by overprovisioning occurring if a scenario is realizing that does not
result in the maximum.
Figure 4.12 lists the cost for installed equipment for a three period plan and various approaches. It provides an overview of how the total cost, which to minimize is the aim of
the optimization, is composed. Value shown are absolute costs in cost units, according
to the NOBEL2 cost model. The category axis is divided in Approach and then further
down to periods. This allows a comparison of costs for individual periods. The first stage
costs are about the same for the multi period approaches. Incremental planning however
has significantly lower first period costs, ten percent well below the others. This suits
the pay-as-you-grow investment approach: first stage Capex are as low as possible. If no
growth takes place money is saved. If growth occurs expansion is more costly but this can
be compensated by increased revenue, caused by the growth.
4.2. UNCERTAINTY IN DEMAND LOAD
31
Absolute cost for uncertainty in demand increase. Example 1
7000,00
6000,00
Cost units
5000,00
4000,00
3000,00
2000,00
1000,00
0,00
1
2
3
4
Scenario
All Periods
Stochastic Programming
Worst Case - All Periods
Incremenal
Figure 4.10: Absolute costs For Uncertain Demand Increase At Known Locations
Detailed cost for uncertainty in demand increase. Example 1
Cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
1
2
3
4
Scenario
All Periods
Stochastic Programming
Worst Case - All Periods
Incremental
Figure 4.11: Costs For Uncertain Demand Increase At Known Locations
CHAPTER 4. RESULTS
32
Cost units
4000
3500
3000
2500
2000
1500
1000
500
0
1
2
All Periods
3
2
3
2
3
All Periods - Worst Case
1
Detailed equipment cost for one scenario
1
Stochastic Programming
Period;
Approach
1
2
Incremental
Figure 4.12: Composition of total cost for various approaches
3
Transponder
Regenerator
OMS
Node
4.3. VARIATIONS IN COST DECREASE
4.3
33
Variations in Cost Decrease
cost(3,1)=(1−0,1)*cost(2,{1,2})
cost(3,1)=(1−0,0)*cost(2,{1,2})
cost(2,{1,2})=(1−0,1)*cost(1)
cost(2,{1,2})=(1−0,1)*cost(1)
cost(3,1)=(1−0,3)*cost(2,{1,2})
cost(3,1)=(1−0,3)*cost(2,{1,2})
cost(1)
cost(1)
cost(3,1)=(1−0,1)*cost(2,{3,4})
cost(3,1)=(1−(−0,1))*cost(2,{3,4})
cost(2,{3,4})=(1−0,3)*cost(1)
cost(2,{3,4})=(1−(−0,1))*cost(1)
Example 1
Example 2
cost(3,1)=(1−0,3)*cost(2,{3,4})
cost(3,1)=(1−0,2)*cost(2,{3,4})
Figure 4.13: Scenarios for uncertainty in cost decrease
Up to this point uncertainty was assumed to influence the demand. However obviously
there are other factors of uncertainty in network planning. One of the additional uncertainties is uncertainty in the development of equipment prices. Emerging new technologies
are lowering prices, however actual values and dates are hard to predict, as developers
usually keep inventions to their selves until the market launch. Additional uncertainty is
due to economical circumstances, benefits from special offers and economy of scale savings
are possible, but it is also possible, the global economic situation causes prices to stagnate instead of fall. An act of nature beyond control may cause temporary or permanent
trouble, thus rocketing prices if highly specialized supply parts are needed.
Additionally it is a goal of this thesis to compare the impact of various uncertainties. Thus
it is interesting to compare the robustness against demand fluctuations with the robustness
concerning variations in equipment prices.
To model realistic cases, scenarios (see Figure 4.13) are chosen to include steep, gentle, or
no decline as well as unpredictable changes together with sudden increase of equipment
cost. The demand is assumed to be known completely to measure only the influences of
the cost decreasing factor.
In Figures 4.15 and 4.14 the results are shown. To maintain the scale used in all other
charts, in Figure 4.14 the cost-bars for incremental are extended in dashed lines. It becomes
obvious, that a very gentle cost decrease, or worse a cost increase, is a disadvantage for
Incremental planning. Generally the true multiperiod approaches SP and both(R, WC)-AP
are very close to the optimal solution, meaning within a one-percent gap. In Figure 4.14
scenario three shows a peak in WC - AP. In scenario 3 we assume that equipment cost
continuously rises, therefore an ill-conceived investment plan relying on savings in late
periods fails badly.
34
CHAPTER 4. RESULTS
Detailed cost for uncertainty in cost development. Example 2
1,15
1,13
Cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
1
2
3
4
Scenario
All Periods
Stochastic Programming
Worst Case - All Periods
Incremental
Figure 4.14: Detailed cost for uncertainty in cost development. Example 2. Dashed bars
in scen. 3 and 4 excessed common upper limit of thesis’ standard chart
Limitations of regarding cost uncertainty with the mentioned models apply due to constraint implying no channels be installed prior to their respective use. Other than that
operational expenditure (opex) would have to be considered in addition to capital expenditure (Capex). This however is not the scope of this thesis.
The probabilities of scenarios should be chosen with respect to economic and market situations. Risk of large expenditures late in the lifetime can be lowered by including possibly
rising costs.
In [MSE08] it is stated that the incremental approach performs better for a higher cost
decreasing factor. Remembering the basic principle of the approach this is quite obvious: as
it is optimizing each period on its own the INC approach delays purchasing of equipment
as long as possible. If the equipment constantly gets cheaper the effect of suboptimal
routing in later periods and the associated need for longer fibers, higher nodal degrees
etc. is compensated by the equipment being cheaper in late stages. Naturally the effect is
reversed with ascending equipment costs worsening the effects of the buy later strategy.
4.3. VARIATIONS IN COST DECREASE
35
Expected cost for uncertainty in cost development. Example 1
Expected cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
All Periods
Stochastic
Programming
Realistic - All Periods
Worst Case - All
Periods
Incremental
Approach
Figure 4.15: Expected cost for uncertainty in cost development. Example 1
36
CHAPTER 4. RESULTS
4.4
Preventing Infeasibilities in Routing
DEM(3,1)=rand(0..10)
DEM(2,{1,2})=rand(0..10)
DEM(3,2)=rand(0..10)
DEM(1)=rand(0..20)
DEM(3,3)=rand(0..10)
DEM(2,{3,4})=rand(0..10)
DEM(3,4)=rand(0..10)
Figure 4.16: Scenarios for provoked blocking
If a steady incline in demand is assumed, and a long planning horizon concerned, a network
can become quite loaded. For those highly loaded networks planning routing ahead of time
is especially important. If routes are chosen only with current demand and minimization
of current costs in mind, blocking may occur quickly. In our model we limit the nodal
degree to ten, and allow only one node at specific nodal site. With respect to the used
US-Network-Topology (4.2) the PA node turns out to be heavily used, lying on one of the
shortest paths for a lot of node pairs. If nodal degree ten is reached, no further demands
can be routed if there is no capacity for OChs left in the already present OMS. This
situation is not only reached, if there are a lot of demands originating or terminating at
the bottleneck node, PA but also if there are a lot of demands between nodes adjacent to
it. In this case the node has to be traversed by OChs between other node pairs, using up
all of PAs capacities.
To model the high load, a uniformly distributed demand over the whole network is assumed.
The number of demands is chosen such that the network is highly loaded, see Figure 4.16.
An initial demand of a uniform distribution of values from zero to twenty is assumed. This
results in feasibility for all approaches. In the following periods additional, again uniform,
distributions of values from zero to ten are added to the total demand. These distributions
are of equal mean demand size, but differ in individual demands between node pairs. Thus
the uncertainty is present, comparable to the Allocation of Demand Uncertainty study
in Section 4.1. Now however no allocation is assumed to be known, as opposed to the
half-knowledge assumed in Section 4.1.
Depending on the actual size and distribution of the demand several cases can be distinguished:
1. No blocking at all: expected high costs for Inc show, performance of Real - AP and
SP comparably and good. This is due to the low actual uncertainty. The scope of
4.4. PREVENTING INFEASIBILITIES IN ROUTING
37
this experiment is to measure quality of approaches for exceptionally highly loaded
networks.
2. Blocking in INC: Incremental approaches work only for a part of the considered
scenarios. Some lead to blocking in late stages as the needed nodal degree exceeds
ten. Real AP, SP do not lead to infeasibilities and result in costs comparably and
efficient, close to the optimum. SP seems to be a little more efficient for those high
demand values. This is due to randomness although minimal, it is present and proves
a handicap for All Periods approaches.
3. Blocking in Real AP: the demand reaches a level that is infeasible by INC approach,
by the second period at the latest, blocking occurs. AP provides the optimal solution
as usual. Real AP is not able to avoid blocking for all cases. SP with its special
characteristic is able to avoid blocking altogether. Costs for SP are close to optimum
due to the large number of installed OCh throughout the network. There simply are
not many alternatives to the given solution.
Further added demand will lead to infeasibilities in SP and at some point not even an AP
plan will succeed.
Costs / feasibilities for very-high-demand scenarios
Cost relative to All Periods cost
1,12
1,10
1,08
1,06
1,04
1,02
1,00
0,00
B
L
O
C
K
I
N
1 G
B
L
O
C
K
I
N
G
B
L
O
C
K
I
N
2 G
B
L
O
C
K
I
N
G
3
B
L
O
C
K
I
N
G
B
L
O
C
K
I
N
4 G
B
L
O
C
K
I
N
G
Scenario
All Periods
Stochastic Programming
Worst Cost - All Periods
Incremental
Figure 4.17: Comparison of approaches’ costs/feasibility for very high demands.
The chosen scenario tree leads to results of above category three. Results are shown
in Figure 4.17. As suggested blocking occurs for every scenario, using the Incremental
planning approach. Stochastic programming leads to good results, close to the optimal All
38
CHAPTER 4. RESULTS
Blocking probabilities
100%
90%
80%
Probability
70%
60%
50%
40%
30%
20%
10%
0%
Stochastic Programming
Realistic - All Periods
Worst Case - All Periods
Incremental
Approach
Routing feasible
Blocking occurs
Figure 4.18: Blocking Probabilities
Periods solution. The same effect was observed for Allocation of Demand Uncertainty in
Section 4.1. Seemingly this kind of uncertainty does not have a highly deteriorating effect
on the costs for planning with Stochastic Programming. Additionally the very high load
of the network does not leave many alternatives for routing, thus leads to results in the
same order of magnitude. The drawback of Worst Case - All Periods is that even though
one scenario is under all circumstances feasible, it turns out to be rather expensive. At
this level of saturation of the network very long routes may have to be taken, resulting in
high costs for unexpected ‘detours’.
In Figure 4.18 again the case of blocking in Real - All Periods is pictured. A probability
of feasibility of 100% for SP is only achievable for certain if the realizations are limited to
the four scenarios considered. Otherwise of course even with SP blocking may occur.
These results are considered the greatest advantage of Stochastic Programming. Even
if Incremental Planning adds flexibility and prevents overprovisioning, it does limit the
number of periods with demand increase. In order to resolve the block in a node, rerouting
has to be done, or additional nodes
Chapter 5
Conclusions & Outlook
Planning is rewarding. Results of the cases studied in this thesis show that even with only
very little certainty about future developments planning ahead can save money. In every
case considered, the costs of Incremental Planning define the upper bound of costs, those
for All Periods the lower bound. The Stochastic Approach leads to results in between. All
Periods costs can not be achieved realistically, the possibilities for approaches with much
better cost effectiveness are limited though, since the Stochastic Programming results are
already very close to the optimum. Often the choice of model used will be limited by what
data is available. As long as the uncertainty does not reach a level where no predictions
can be made at all the Stochastic Planning approach can be used to increase robustness of
long term plans. If no single assumption about the future can be made it is questionable
if an investment should be made after all, since it resembles a gambling game more, than
it is a sensible economic decision.
As a foundation for planning a network in the real world the model for demand uncertainty
and cost uncertainty should be combined into one to offer the most possibilities to include
uncertainty. Since this thesis only included a Sensitivity Analysis, as a next step Heuristics
must be developed and evaluated, after that simulations will give statistical properties.
Only then the real performance of the Stochastic Programming approach can be expressed
in numbers.
Still then decisions must be made wisely, all approaches presented can only result in
planning-tools. Also those approaches have to be chosen according to and possibly be
adapted to special environments. Using the wrong tool can result in a faulty decision. In
the whole decision process, the economic environment plays a major role, cost considerations cannot be made without analysis of the market situation. Present competitors and
possible partners for cooperation must be identified, as greenfield solutions are rather rare
nowadays.
Appendix A
Abbreviations
CDF
Capex
DCF
DGE
Ampl
OLA
LP
ILP
OMS
OCh
EOL
INC
AP
SP
WC-AP
RO
RWA
WA
WDM
Cost Decreasing Factor
Capital Expenditures
Dispersion Compensating Fiber
dynamic gain equalizer
A Mathematical Programming Language
Optical Line Amplifier
Linear Programming or Linear Program
(Mixed) Integer Linear Programming or Integer Linear Program
Optical Multiplex Section
Optical Channel
End Of Life Planning
Incremental Planning
All Periods Planning
Stochastic Programming
Worst Case All Periods Planning
Robust Optimization
Routing and Wavelength Assignment
Wavelength Assignment
Wavelength Division Multiplexing
Table A.1: List of Abbreviations
List of Figures
2.1
Illustration of components needed to connect two nodes with one Optical
Channel on one Optical Multiplex Section . . . . . . . . . . . . . . . . . .
6
3.1
3.2
Sample scenario tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of results given by Stochastic Programming model . . . . . . .
14
20
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
Overview of Cost-Dependency on Factor of Uncertainty . . . . . . .
US Topology used in models . . . . . . . . . . . . . . . . . . . . . .
Scenario - Location (1) . . . . . . . . . . . . . . . . . . . . . . . . .
Expected Costs - Location (1) . . . . . . . . . . . . . . . . . . . . .
Scenario Costs - Location (1) . . . . . . . . . . . . . . . . . . . . .
Scenario Costs - Location (2) . . . . . . . . . . . . . . . . . . . . .
Scenarios for uncertainty in demand increase . . . . . . . . . . . . .
Expected Costs - Increase (1) . . . . . . . . . . . . . . . . . . . . .
Expected Costs - Increase (2) . . . . . . . . . . . . . . . . . . . . .
Absolute Scenario Costs - Increase (1) . . . . . . . . . . . . . . . .
Scenario Costs - Increase (1) . . . . . . . . . . . . . . . . . . . . . .
Composition of total cost for various approaches . . . . . . . . . . .
Scenarios for uncertainty in cost decrease . . . . . . . . . . . . . . .
Scenario Costs - Cost Decreasing (2) . . . . . . . . . . . . . . . . .
Expected Costs - Cost Decreasing (1) . . . . . . . . . . . . . . . . .
Scenarios for provoked blocking . . . . . . . . . . . . . . . . . . . .
Comparison of approaches’ costs/feasibility for very high demands.
Blocking Probabilities . . . . . . . . . . . . . . . . . . . . . . . . .
22
23
24
25
26
27
28
29
29
31
31
32
33
34
35
36
37
38
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List of Tables
2.1
2.2
2.3
Relative cost values provided by NOBEL 2 multilayer cost model . . . . .
Different Multiperiod Approaches Used in This Thesis . . . . . . . . . . . .
Levels of Uncertainty According to Grover [LG05] . . . . . . . . . . . . . .
5
8
11
A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
45
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