Bachelor's Thesis: Demand Uncertainty in Multiperiod...

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Technische Universit¨ at M¨ unchen Lehrstuhl f¨ ur Kommunikationsnetze Prof. Dr.-Ing. J¨ org Ebersp¨ acher Bachelor’s Thesis Demand Uncertainty in Multiperiod Network Planning Author: Thilo Sch¨ ondienst Matriculation Number: 2838050 Address: Christoph-Probst-Str. 12 80805 M¨ unchen Germany Email Address: [email protected] Supervisor: Clara Meusburger Begin: 01.09.2009 End: 11.12.2009

Transcript of Bachelor's Thesis: Demand Uncertainty in Multiperiod...

Technische Universitat MunchenLehrstuhl fur Kommunikationsnetze

Prof. Dr.-Ing. Jorg Eberspacher

Bachelor’s Thesis

Demand Uncertainty in Multiperiod Network Planning

Author: Thilo SchondienstMatriculation Number: 2838050Address: Christoph-Probst-Str. 12

80805 MunchenGermany

Email Address: [email protected]: Clara MeusburgerBegin: 01.09.2009End: 11.12.2009

Abstract

In multiperiod planning for optical networks, future developments of demand, cost, andother planning parameters are considered, in order to decide upon when and where toimplement and upgrade network equipment. Thus, with perfect knowledge of the future acost optimal solution can be achieved. Evidently though, the future is uncertain. In thisthesis an approach known as stochastic programming is used to increase robustness againstrandom fluctuations in demand and equipment cost. Several future scenarios are consid-ered and weighted with probabilities. It is shown that by using stochastic programmingthe robustness of multiperiod network planning concerning different types of uncertainty(demand allocation, demand increase, development of equipment cost) can be increasedand infeasibilities in routing can be prevented.

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Kurzfassung

Bei der Planung optischer Netze uber mehrere Perioden hinweg, werden die zukunftigenEntwicklungen von Verkehrsaufkommen, Kostenentwicklung und weiterer Parameterberucksichtigt, um zu entscheiden, wann, wo und welche Netzwerkkomponenten instal-liert oder erweitert werden. So kann bei genauer Kenntnis der Zukunft eine, in Hinblickauf die Kosten, optimale Losung berechnet werden. Die Zukunft jedoch ist ungewiss. Indieser Arbeit wird die Stochastic-Programming-Methode verwendet, um die Robustheit derPlanung gegenuber Schwankungen in Verkehrsanforderungen und Komponentenkosten zuerhohen. Dazu werden mehrere Szenarien fur die zukunftige Entwicklung erstellt und mitWahrscheinlichkeiten gewichtet. Es wird gezeigt, wie unter Einbeziehung der stochasti-schen Eigenschaften von Zukunftsvorhersagen, die Robustheit der Netzplanung gegenuberverschiedenen unsicheren Faktoren (ortliche Verkehrsunsicherheit, Unsicherheit im Anstiegdes Verkehrsaufkommens, Unsicherheit bei den Kosten der Netzkomponenten) erhoht wer-den kann. Desweiteren konnen Blockaden in der Verkehrslenkung, ausgelost durch hohenAnstieg des Verkehrs, durch vorausschauende Planung vermieden werden.

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Contents

1 Introduction 1

2 State of the Art 32.1 Multiperiod Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Decision making Under Demand Uncertainty . . . . . . . . . . . . . . . . . 10

2.2.1 Stochastic Programming . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Modeling of Uncertainty in Multi Period Planning 133.1 Representation of Uncertain Development . . . . . . . . . . . . . . . . . . 133.2 Stochastic Programming Model for Uncertain Demand . . . . . . . . . . . 153.3 Stochastic Programming Model for Uncertain Cost . . . . . . . . . . . . . 183.4 Structure of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Results 214.1 Allocation Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Uncertainty in Demand Load . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Variations in Cost Decrease . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Preventing Infeasibilities in Routing . . . . . . . . . . . . . . . . . . . . . . 36

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. Toprovide both for all of the network’s lifetime it is necessary to plan ahead. Specifically, largescale backbone networks come with a high amount of capital expenditures (Capex), aimingto 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 risedue 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 multiperiod planning. Moreover, the uncertain nature of these changes can be accounted for byapplying stochastics to the planning.

Minimizing costs while maximizing operability can be stated as an optimization problemas follows: For every demand requirement between two nodes an appropriate route for thelightpath from the source node to the target node has to be identified. Depending on thisrouting the equipment (nodes, transponders, amplifiers, and regenerators) is placed in thenetwork [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 ofpenalty cost for blocking or capacity leasing [Leu05], [VCPD07] are not addressed. Theseapproaches would result in reducing the importance of feasible routing, focussing on costefficiency.

This thesis’ aim is to include Stochastic Programming into already implemented multiperiod approaches. The motivation is to increase robustness of planning, that is to reducethe negative effects of stochastic changes in demand and other parameters. Another aspectof interest is what kinds of uncertainty there are, that is which parameters have randomcharacteristics. A differentiation is made, how severe the impact of different uncertainfactors is.

This thesis is organized as follows: Chapter 2 gives an introduction to the ILP formulationof 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 uncertaintyis given in Section 2.2. The Stochastic Programming approach is introduced in Subsec-tion 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 modelsand 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 IntegerLinear Programming (ILP)[Leu05]. This term refers to an optimization problem with alinear objective function subject to linear constraints.

Maximize cTxSubject 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 orminimized is called the objective function (cTx in this case). The equations Ax ≤ b arethe constraints which specify a convex polytope over which the objective function is to beoptimized.

The solution of the optimization has to consist of integer values, due to the fact that apiece of network equipment can only be installed as a whole. Simply calculating resultsallowing 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 nec-essary to express the problem in a mathematical way first. It is then transformed into aprogram written in ‘A Math Programming Language’ (AMPL). AMPL is a suitable lan-guage because of its syntax being very close to a mathematical notation. In addition thewidely used CPLEX solver is used to do the actual optimization. It employs Branch andBound algorithms to solve ILP problems [IBM09].

The network is modeled as a undirected graph G = (V,E) where V is the set of nodes andE 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′} : v, v′ ∈ V, v 6= v′}

is designed.

A set of predefined paths

Pnp,p ⊂ E∀np ∈ N∀p ∈ {1, . . . , noOfPathsnp}

is defined where the parameter p gives the number of a certain path between the node pairand is an element of the set {1, . . . , noOfPathsnp}. To maintain a reasonable simulationtime, this set of paths considered is a subset of paths containing only the ten shortestpossible ones for each nodepair.

An Optical Channel OChnp,p indicates the channel between nodes np alongside path num-ber p. One OCh is needed to transport a demand of value one, specified in the demandmatrix. An Optical Multiplex Section OMSe denotes one installed optical multiplex sec-tion on edge e. This system is capable of connection nodes. It must be installed alongsidethe path an OCh takes. One OMS can be used by multiple OChs depending on thecapacity of the equipment used.

The network model is now completely defined. To do an optimization however, an objectivefunction and conditions and must be defined. The goal of the optimization is to calculatethe lowest possible cost for the network, hence it is called the cost function. We assignprices 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 costsof a 10Gbit/s transponder with a maximum transmission length of 750 km. The compo-nents available are bidirectional, capable of 80 wavelengths. An overview of equipmentcosts is given in Table 2.1.

In our model each optical channel needs two transponders (with cost ct). If the length ofthe channel exceeds the transparent transmission length, regenerators (with cost cr) areinstalled. The length of a channel is the sum of the traversed edge lengths plus a penaltyfor every transparent node passed through. Each installed OMS has fiber costs cfibercontaining costs for components: Optical Line Amplifier (OLA), dispersion compensatingfiber (DCF), and the dynamic gain equalizer (DGE).

DCF and Optical Line Amplifier are installed in the same interval on the fiber. A Dynamicgain equalizer is needed at every fourth OLA site. The node costs cnode depend on therequired nodal degree. The nodal degree equals the number of OMS connected to a certainnode and cannot exceed the degree of 10. The number of installed nodes is limited to onenode per nodal site.

2.1. MULTIPERIOD PLANNING 5

Equipment Abbr. in costfunction

Reach Cost valuerelative to 10Gtransponderwith 750kmreach

Optical LineAmplifier

cfibere 1500 km 3000km

2.77 3.45

Dispersioncompensationfiber

cfibere 1500 km 3000km

0.728 0.88

Dynamic gainequalizer

cfibere 3.17

Nodal switch d= 1

cnoded 10.83

Nodal switch d= 2

cnoded 25.30

Nodal switch d= 3, 4, 5

cnoded 10.42 x d +2.75

Nodal switch d= 6, ..,10

cnoded 11.11 x d +2.75

10G transpon-der

ct10G 1500 km 3000km

1.25 1.67

10G regenera-tor

cr10G 1500 km 3000km

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, connectedby 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 DispersionCompensation 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 toevolve, enabling higher nodal degrees in future.

The capacity constraint 2.1 limits the number of OChs per edge to the maximum numberof wavelengths times the number of installed OMS.

∑np,pnp:e∈P

OChnp,p ≤ noOfWavelengths ×OMSe ∀e ∈ EDGES (2.1)

The demand transported constraint 2.2 ensures an OCh is present for every demand entry

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Figure 2.1: Illustration of components needed to connect two nodes with one OpticalChannel on one Optical Multiplex Section

in the demand matrix.

∑pnp

OChnp,p = dnp ∀np ∈ N (2.2)

The objective function to minimize is the cost function Cost . It is composed of the sumof the cost of each part of equipment needed for the solution imposed by the constraints.

Cost =∑e

OMSe × cfibere

+∑np,pnp

OChnp,p × ctnp,p × 2

+∑np,pnp

OChnp,p × crnp,p × noOfRegeneratornp,p

+∑d,v

noded,v × cnoded (2.3)

If an integer solution is found for the optimization problem, the values we are interestedin are OMSe, OChnp,p and the resulting nodal degree of each node. These results fullydefine the placement of equipment needed to satisfy the demand.

The optimization is written in single period notation, the planning results in a networksatisfying 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 amountof time, for example a year. It denotes the time that passes between upgrading of networkequipment to satisfy changing demand.

We limit the optimization to routing and skip wavelength assignment to save calculationtime. Complexity is greatly reduced by leaving out the assignment of colors to OChs. Theremaining routing problem still has rather long runtimes.

Depending on the chosen approach (see Table 2.2), all demand matrices are provided inadvance or sequentially one period of time after the other. Optimization software thenfirst calculates if the problem is feasible with the input data given or not. If the problemis feasible a non–integer solution is calculated, giving a lower bound for the solution.Iteratively integer–solutions are computed until optimality is achieved. Conveniently forthe solver a gap criterion, expressed as a percentage, may be defined. If a solution is foundwithin this percentage around optimality the iterations are stopped to save simulationtime.

The optimization process is done, according to the chosen approach, once for all periodstogether (for approaches All Periods and Stochastic Programming), incrementally one pe-riod at a time (for the Incremental approach). In a combination of the two calculation isdone repeatedly for all periods together every time a change of the input data occurs (AllPeriods with Deviating Demand approach). If the optimization is done incrementally andinfeasibility occurs a time after period one, blocking occurs, demands can not be routedand planning has failed.

In the model developed, one time period can equal any chosen amount of time, for exampleyear. Shorter durations may increase flexibility and a planning at mid-year, or even shortermay be established.

During the planning horizon, as mentioned earlier, parameters vary. Observations haveshown that equipment prices are generally gradually reduced, due to new technologiesemerging. Therefore we use a declining-cost model with start values from [HGMS08] anda cost decreasing factor (cdf). Depending on approach and studied effect of uncertaintyon 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 includedin this thesis.

Table 2.2 shows some characteristics of different approaches used in this study. TheILP code for Incremental and All Periods approaches has been developed in connectionwith [MSE08]. Using the Incremental, the cost function is optimized for one period afteranother. The expenses per period are hence the lowest possible regarded individually. Thefuture is not concerned in planning but only the current periods parameters: the Incre-mental approach is no true multi period approach. The All Periods approach optimizesthe cost function for all periods, for which forecasts are available at once, leading to anoptimal 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 forthe network planning. Running an optimizer on the cost function, costoptimality is achieved for a single time period. However the fact thatpreviously installed channels remain fixed in place, leads to an overallsolution (concerning more than one period of time) that is typicallysuboptimal.

All Periods planning:The All Periods approach minimizes network costs over multiple pe-riods of time for given forecasts. Demand matrices are estimated inadvance, one for each time step, e.g., a year. Parameters of the costfunction are assigned additional indices t for time. The optimizationis done for all t ∈ noOfPeriods at once; hence the solution is optimalfor 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 worldsetting however the actual demand is unlikely to be known a couple oftime steps in advance. Accepting this lack of knowledge the approachis revised; after each time step, the demand estimated in advance isreplaced with the realized actual demand. For the corrected values anoptimal solution for the rest of the lifetime is calculated. This revisionstep is done after each period. Additionally the remaining forecastsmay be replaced with newer ones, to achieve better solutions.

Stochastic Programming: (see Chapter 3)The uncertainty of future development is taken into account by as-suming not one but a set of possible evolutions is known. A solutionfor the deployment and routing is calculated that minimizes the totalexpected 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 solutionsof the previous planning are fixed. Then the input Values for Demand are changed andbased upon them a new All Periods solution is calculated. This procedure is repeated everyplanning 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 planningconditions 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 theinclusion of uncertainty to further research. [AKP03] discusses uncertainty in capacityexpansion 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 noknowledge of demands. Cost efficiency however is not concerned. [LZS05] minimizes hopcount with only partial knowledge of demand, using load balancing. However multi periodand budget considerations are not made. [KLO+03] uses stochastic programming to tacklerouting under uncertainty for a given budget. Long term planning however is not addressed.[AZ07] extends single-stage robust optimization to two stages. Multi period considerationsare 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 informationon trends of future traffic behavior and plan the network accordingly. [MW05] proposesa stochastic approach to maximize revenue under uncertain demand conditions. Capexminimization for multi period investments is no concern, though. [ZMLB08] studies theimpact of uncertainty in physical parameters on dimensioning of optical networks.

In the proposed Stochastic Programming Multi Period approach cost efficient multi periodplanning 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 theexact layout of how to build or upgrade a network.

According to [BHS99] a good decision is based on logic, it considers all available data andpossible 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 basedon 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 insuccessful outcomes in the long run.

“For every decision, the steps that need to be taken in order to make it a good decisionare 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 decisionsis known for sure. Every realization of a state of the world sj is fully defined. Thealternative, 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 realizingstates sj are known, their probabilities however are not.

For decision under risk, expected values of consequences of all decisions can be calculatedeasily since all probabilities are given. Decision under ambiguity can be addressed withthe principle of indifference. The principle states, that if no knowledge is available aboutthe probabilities of outcomes, and there is no knowledge indicating unequal probabilities(John Maynard Keynes), every possibility is assigned the same probability of 1

N: (N =

number of possible outcomes). With this assumption made, the expected value can be

2.2. DECISION MAKING UNDER DEMAND UNCERTAINTY 11

Level I:A Clear-EnoughFuture

A single forecast is suffices for network planning. Uncer-tainty is neglectable, cost and effort to include it wouldoutsize the possible gain. This was valid for example inpast telephone networks.

Level II:AlternativeFutures

A handful of alternatives can be identified. These dis-crete scenarios are assigned probabilities, possibly noteasy to quantify.

Level III:A Range ofFutures

Here no distinguishable discrete scenarios can be given.The alternatives are countless, and a large number ofscenarios merely define some boundaries.

Level IV:TrueAmbiguity

No reasonable prediction can be made in this case. Pos-sibly 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 expectedvalue for optimization. It can therefore be applied in both cases.

A slightly different approach by [LG05] divides uncertainty in four levels(Table 2.3). LevelI clearly belonging to decision under certainty and Level II meaning a decision underrisk. Uncertainty of Level III still allows a decision to be made applying the principle ofindifference. Level IV however implies that the decision can not be made on assumptionsabout the future but must be made independently. An approach to do network planningunder these conditions is presented in [Sch06].

Once the environment has been identified, the appropriate mathematical decision modelhas to be chosen. Suitable models for Decision Making Under Risk are, for example EMV(expected monetary value), EVPI (expected value of perfect information), EOL (expectedopportunity loss), and Sensitivity Analysis. All these are based on some kind of expectedvalue calculation. If the environment proves to be ambiguous appropriate models includeMaximax, Maximin, and Minimax.

As economical decision theory lies beyond the scope of this thesis, further investigationson mathematical decision models are not made.

In the global market economy a variety of additional methods can be used to simulatea network construction project. If other forces influencing parameters become involved,the theoretical field changes from stochastics towards game theory. Here, other playersare included and simulation becomes much more complex. Influences can come from com-petitors, contractors, consumers, and customers. In [VCPD07] an approach utilizing RealOptions is explained in detail. This approach involves someone willing to grant the optionsand a revenue from sold or let capacity. Regarding flexibility a high gain is possible fromhaving 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 troughoverprovisioning. The Stochastic Programming approach developed in this thesis may wellbe used to calculate the price of an option, or to give the value of an insurance, coveringfor penalty expenses.

2.2.1 Stochastic Programming

Deterministic optimization models do not represent reality. However a decision makingtool is of little value if the underlying simulation is unrealistic. One well known way ofdealing with optimization problems under uncertainty is Stochastic Programming [KW94].Multiple disciplines take advantage of this approach. Stochastic Programming for examplebenefits financial applications like portfolio optimization or applications from operationalresearch like fleet management or inventory problems [van07].

The interpretations of Stochastic Programming are manifold. Generally speaking it meansthe inclusion of stochastic parameters in a linear program. Hence, Stochastic Programmingprovides means to include alternative future outlooks and to weight each with a probabilityof occurrence, thus it approximates reality much better than deterministic approaches.

Linear programs are problems that can be expressed in canonical form:

Maximize cTxSubject 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 orminimized is called the objective function (cTx in this case). The equations Ax ≤ b arethe constraints which specify a convex polytope over which the objective function is to beoptimized.

The conversion to a stochastic program can be done by simply adding a random vector ξto the constraint: Ax ≤ b + ξ.

Consequently the goal of the optimization changes to the expected value E(cTx)

Thus the Stochastic Program is:

Maximize E(cTx)Subject to Ax ≤ b + ξ.

Chapter 3

Modeling of Uncertainty in MultiPeriod Planning

The goal of multiperiod planning is to deal with time dependent planning parametersin a cost efficient way. Throughout the operation time of a network several parameterswhich are influencing the optimal planning solution are changing. A single period planningapproach can not address these changes which are occurring at some point in networklifetime. More efficient solutions can be examined by means of multiperiod planning. Inprevious studies the development of parameters over time has been assumed to be known forsure. These have shown interesting insights into network planning with cost minimizationin mind [MSE08]. As mentioned in [MSE08] [SKS06] further challenges of multiperiodapproaches lie in the parameters’ unpredictable nature. While the current situation isknown to a high degree, statements about future developments can never be made withcertainty. The contribution of this thesis is to include the uncertainty of parameters intomulti period planning models.

3.1 Representation of Uncertain Development

To be able to deal with parameter development in a mathematical way a representation isneeded 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 uncertaintyhas to be reduced to a level where alternatives can be specified and their respective prob-abilities be estimated. These different development forecasts are called scenarios. Wheredifferent scenarios are concerned a widely used representation is a tree shape as picturedin 3.1 called Scenario Tree [AKP03] . The widening tree structure with a number of nodesincreasing with time represents the assumption that reality is known fairly accurate in the

14 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING

1

22

21

31.1

31.2

32.1

32.2

1Time

Planning forPeriod 1

Planning forPeriod 2

Planning forPeriod 3

Figure 3.1: Sample scenario tree spanning three periods depicting four scenarios; scenario1 is highlighted

near future and less further away. Also a dependency of developments on previous ones isconsidered a realistic model [DCW00].

For the scenario trees used in this thesis we assume the structure given in Figure 3.1. Eachnode represents a possible state. In the case of demand uncertainty this means for each nodeN(p) with p ≥ 1 a demand matrix DEM(p, s) with (p ∈ PERIODS, s ∈ SCENARIOS)is given. If uncertainty in the decrease of equipment cost is regarded, for each node a costdecreasing factor cdf(p, s) with (p ∈ PERIODS, s ∈ SCENARIOS) is needed. Planningis done at time zero. The first period contains only one node since the initial state ofthe network is known. Future periods contain 2(p−1) alternative nodes. This values arechosen to create scenarios that can easily be differentiated and to reduce simulation time byreducing complexity. For a closer approximation more nodes, and more scenarios may beincluded, this does however not change the observations made in this thesis. It leads to anumber of possible outcomes that increases with time. Therefore we act on the assumptionof a more uncertain development the further a prediction goes.

Leaf-nodes, i.e., nodes without emerging edges, can be reached by only one single possiblepath 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 scenariosare always numbered from top of the tree to the bottom in this thesis. Three periodsare given with four leaf nodes, that is four alternatives in the final stage. Therefore fourscenarios 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 thedevelopment of demand, the other being uncertainty in the development of equipment costmodeled by the common cost decreasing factor.

3.2 Stochastic Programming Model for Uncertain De-

mand

A popular interpretation of stochastic programming for multi period simulations is a twostage model, meaning two periods are covered by the approach. Here an immediate problemis solved, with knowledge about possible futures in mind. For the possible futures alreadyappropriate reactions are given. Depending on the actual realization of the future, thesegiven solutions have to be applied. We adapt this idea and extend it to a three stages, orthree periods.

However the special property is that by means of nonaticipativity constraints a single so-lution is found for the first stage. This single solution prepares the path well for anypossible future, optimizing the expected value of the outcome. A commonly used repre-sentation of these alternative developments are scenario trees [DCW00] see Figure 3.1 foran 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 Periodsvariant of the model, that optimizes the cost function for all considered time periods atonce. This way all the time dependent parameters and variables were already in possessionof 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 oneto cover multiple scenarios. The variable s denotes a scenario)

DEM(t, s) = [dnp]np×np t ∈ PERIODS, s ∈ SCENARIOS (3.1)

The input data for the demand is extended by one dimension representing the sce-narios. A demand matrix is created for each scenario in each period. However dueto the nature of the chosen scenario tree, in the first period demand matrices for allscenarios are equal. In the following stages the number of unique matrices equals tothe number of nodes in the tree at that stage. (e.g. in a model of 3 periods with fouralternative 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 ∈ SCENARIOS) (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 solutionsfor all scenarios

OMS(p, s, edge) ≥ 0 integer (3.3)

OCh(p, s, np, subset paths) ≥ 0 integer (3.4)

node(p, s, d, n) binary (3.5)

4. Adapt existing constraints to be imposed on each scenario

Capacity constraint:

∑np,pnp:e∈P

OChnp,p,t,s ≤ noOfWavelengths ×OMSe(t, s)

∀e ∈ EDGES∀t ∈ PERIODS∀sinSCENARIOS

Demand transported constraint:

subset paths(np)∑pathNo

(OCh(p, t, s, np, pathNo)) ≥ dem(t, s, np)

∀t ∈ PERIODS, s ∈ SCENARIOS, np ∈ DEMANDPAIRS, p ∈ PATHS

5. Add nonanticipativity constraints

3.2. STOCHASTIC PROGRAMMING MODEL FOR UNCERTAIN DEMAND 17

OMS(1, s, edge) = OMS(1, s+ 1, edge)

(∀edge ∈ EDGES, s ∈ SCENARIOS)

∀edge ∈ EDGES : OMS(2, 1, edge) = OMS(2, 2, edge)

∀edge ∈ EDGES : OMS(2, 3, edge) = OMS(2, 4, edge)

OCh(1, s, np, pathNo) = OCh(1, s+ 1, np, pathNo)

(∀s ∈ (SCENARIOS − 1), np ∈ DEMANDPAIRS, pathNo ∈ subsetpaths(np)) :

OCh(2, 1, np, pathNo) = OCh(2, 2, np, pathNo)

(∀np ∈ DEMANDPAIRS, pathNo ∈ subsetpaths(np)) :

OCh(2, 3, np, pathNo) = OCh(2, 4, np, pathNo)

(∀np ∈ DEMANDPAIRS, pathNo ∈ subsetpaths(np)) :

node(1, s, d, n) = node(1, s+ 1, d, n)

(∀s ∈ (SCENARIOS − 1), n ∈ NODES, d ∈ DEGREE)

node(2, 1, d, n) = node(2, 2, d, n)

(∀n ∈ NODES, d ∈ DEGREE)

node(2, 3, d, n) = node(2, 4, d, n)

(∀n ∈ NODES, d ∈ DEGREE) (3.6)

The above are called nonanticipativity constraints. These make sure the result of theoptimization is not four different strategies but a single one for the next step andresembling the scenario tree thereafter.

If more than tree periods with four scenarios having the given tree-structure areconsidered these constraints have to be changed accordingly. The Constraints forperiod two are only applicable to the chosen tree structure with two possible statesin 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 thiscase the nonanticipativity constraint for the first period is crucial, as one decision isneeded 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

minE(costs) :

NS∑s

p(s) ∗NP∑p

EDGES∑e

OMS(p, s, e) ∗ costfiber(p, e)

+DEMANDPAIRS∑

np

subsetpaths(np)∑pathNo

OCh(p, s, np, pathNo) ∗ 2 ∗ costtrans(p, np, pathNo)

+DEMANDPAIRS∑

np

subsetpaths(np)∑pathNo

OCh(p, s, np, pathNo) ∗ costregen(p, np, pathNo)

+NODES∑

n

DEGREE∑d

node(p, s, d, n) ∗ costnode(p, d)−node(p− 1, s, d, n) ∗ costnode(p, 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 scenariosweighted with their probabilities

3.3 Stochastic Programming Model for Uncertain

Cost

The above steps were taken but instead of indexing the parameters dependent on demandover scenarios anything related to cost was indexed. Concerning the marked (†) steps theexisting 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 ∈ PERIODS × s ∈ SCENARIOS] (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 thecircumstance that costs may develop differently.

3.4. STRUCTURE OF RESULTS 19

costnode(p ∈ PERIODS, s ∈ SCENARIOS, d ∈ DEGREE)

costtrans(p ∈ PERIODS, s ∈ SCENARIOS, np ∈ DEMANDPAIRS, pathNo ∈ subset paths(np))costregen(p ∈ PERIODS, s ∈ SCENARIOS, np ∈ DEMANDPAIRS, pathNo ∈ subset paths(np))

costfiber(p ∈ PERIODS, s ∈ SCENARIOS, edge ∈ EDGES) (3.9)

7. Change objective function to be the expected value of the total cost

minE(costs) :

SCENARIOS∑s

p(s) ∗PERIODS∑

p

EDGES∑e

OMS(p, s, e) ∗ costfiber(p, s, e)

+DEMANDPAIRS∑

np

subsetpaths(np)∑pathNo

OCh(p, s, np, pathNo) ∗ 2 ∗ costtrans(p, s, np, pathNo)

+DEMANDPAIRS∑

np

subsetpaths(np)∑pathNo

OCh(p, s, np, pathNo) ∗ costregen(p, s, np, pathNo)

+NODES∑

n

DEGREE∑d

node(p, s, d, n)∗costnode(p, s, d)−node(p−1, s, d, n)∗costnode(p, s, d)

(3.10)

Note the indices at the cost parameters.

For the inclusion of various cost scenarios in the Incremental approach only minormodifications were necessary, since equipment costs were already calculated period-ically inside a loop. A parameter which cdf to use in which period was all that hadto 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 StochasticProgram. In this thesis however only initial greenfield planning is considered, giving thedecisions for the first three periods, based on four initially forecasted demand scenarios.

20 CHAPTER 3. MODELING OF UNCERTAINTY IN MULTI PERIOD PLANNING

OMS1; OCh

1; node

1

OMS2.1

; OCh2.1

; node2.1

OMS2.2

; OCh2.2

; node2.2

OMS3.1.1

; OCh3.1.1

; node3.1.1

OMS3.1.2

; OCh3.1.2

; node3.1.2

OMS3.2.1

; OCh3.2.1

; node3.2.1

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 beused without modification, since it already is the expected value. For previously existingapproaches, i.e., Incremental and All Periods, it is necessary to calculate the costs scenario-wise first and then use E(costs) =

∑SCENARIOSs (p(s) ∗ costs(s)) to calculate the expected

value.

Chapter 4

Results

To compare the Stochastic Programming approach to the previously existing, several casestudies have been conducted. Additionally the influence of various kinds of uncertaintyon variations in total cost are of interest. It is shown that depending on the factor ofuncertainty the potential cost savings from multi period planning may vary. To studythe impact of various parameters of uncertainty, each of them has been isolated to allowindividual studies. Those factors are Uncertainty in Allocation of Demand, Uncertaintyin Increase of Demand and Uncertainty in Cost Development. Finally the network wasassumed to be highly loaded and it was tested whether the approaches led to results atall. These sensitivity analysis offer a rough estimation of the performance of the stochasticprogramming approach.

In Figure 4.1 an overview of the results is shown. The heights of the bars indicate the meanexpected cost of an approach relative to the optimum. For the expected costs calculationequal likelihood of any scenarios to realize is assumed. The data used is the mean expectedvalue of all studies conducted for the particular uncertain parameter. The categories arethe afore mentioned three factors of uncertainty, explained in the following sections of thischapter. Measures for Incremental and Stochastic Programming approach are the expectedcosts calculated as explained in Chapters 2 and 3, respectively. All-Periods alway resultsin one as it is equal to perfect knowledge of the scenario realizing and therefore gives thecost optimal planning solution. This can never be realistically achieved since there is noperfect knowledge of the future. It does however give the lower bound of costs. To the All-Periods with Deviating Demand Approach two different measures are applied: RealisticAll-Periods considers planning for every of the four scenarios with realization of every ofthe four scenarios. Using the All Periods with deviating demand approach (Table 2.2), thisresults in sixteen possible cost results. In the chart, the expected value is given, assumingthat all combinations of forecasted/realized scenario have the same likelihood. Worst CaseAll-Periods considers again all combinations of scenarios predicted/realized. Thus, for theresulting costs of each realized scenario only the forecast most differing in total cost isconsidered, that is the worst case is assumed to happen. The mean considers the four

22 CHAPTER 4. RESULTS

Expected cost for various uncertain parameters

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

Demand allocation Demand increase Cost decrease

Uncertain parameter

Exp

ecte

d c

ost

rel

ativ

e to

All

Per

iod

s co

st

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 isobvious. All other approaches have expected costs closer to the optimum. Thus planningahead generally promises to be rewarding in terms of total expected cost. For Uncer-tainty in Demand Allocation and for Uncertainty in Cost Decrease the true multi periodapproaches perform all very well, within one percent from the optimum. ConsideringUncertainty in Demand Increase the expected costs for these approaches are higher. In-cremental planning does, however perform better than in the other cases of uncertaintyconsidered. This effect is due to overprovisioning, meaning excess capacity is provided. Indetail this is explained in Section 4.2.

The data used modeling the network is the USA-NSF-Net taken from [OPTW07] picturedin Figure 4.2. Input demand values were modeled according to the scope of the particularstudy. A steady increase in demand was assumed.

23

~ 1

000

km

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Sea

ttle

CA

1

Pal

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CA

2

San

Die

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UT

Sal

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NE

Linc

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24 CHAPTER 4. RESULTS

4.1 Allocation Uncertainty

In the first case studied, the allocation of the demand is assumed to be uncertain. Theparameters 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 halfis randomly distributed. This is done to achieve comparable results. If the allocation wascompletely unknown results would differ extremely for each run, randomizing the completedistribution.

Period 2

DEM(2,{3,4}

DEM(2,{1,2})

DEM(2,{1..4})

Period 1

DEM(1,{1..4})

Period 3

DEM(3,{1..4})

DEM(3,4)

DEM(3,3)

DEM(3,2)

DEM(3,1)

++

++

+

+

Figure 4.3: Demand tree for allocation uncertainty

The demand tree is composed as in Figure 4.3. Only the additional demand per period isshown 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 thesecond period the common allocation of a base demand (yellowish) is known. This basedemand is the same for both possible developments. Equally for period three demands onecommon base demand for all four possibilities is assumed. On top of the base demandsan individual demand matrix, that about equals the base demand in terms of volume, notallocation 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 toa node already covered by base demand.

The values of demand are chosen in a way that no blocking occurs for any chosen approachand a reasonable maximum nodal degree is reached. Which in our example is equal to aninitial demand of uniformly distributed values from zero to eight. In the following periodsthe added demand is the sum of two uniform random distributions, the common basedemand and the additional individual demand. In period two both distributions assumevalues from zero to two resulting in an overall distribution between zero and four, in thethird and last period they both range from zero to one. Summed up these values result inthe 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

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

All Periods StochasticProgramming

Realistic - All Periods Worst Case - AllPeriods

Incremenal

Approach

Exp

ecte

d c

ost

rel

ativ

e to

All

Per

iod

s co

st

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 ofbars equals the expected value of the results of the approaches used relative to the expectedvalue for All Periods, assuming equal likelihood of scenarios to realize. In Figure 4.5 thecost for the individual scenarios is compared. The cost is expressed relative to the cost forthe optimal All Periods approach in all charts.

In Figure 4.4 it is obvious that the Stochastic Programming and the Worst Case All-Periods 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 multiperiodapproaches. The Incremental approach results in expected costs roughly 10% above theoptimum.

26 CHAPTER 4. RESULTS

Detailed cost for uncertainty in demand allocation. Example 1

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 2 3 4

Scenario

Co

st r

elat

ive

to A

ll P

erio

ds

cost

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, representingthe third scenario, is elevated above those for Stochastic Programming in any other sce-nario. While outperformed by Stochastic Programming in scenarios one, two, and threeWorst Case All-Periods is (though the differences are small) better in three. This resultis due to the fact that on further examination the randomly generated matrices for thirdperiod demand in scenarios 1, 2, and 4 turned out to be highly correlated and quite differ-ent 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 thenetwork components is calculated in favor of these as they yield a higher combined prob-ability. In case it turns out scenario three equals the actual development, third periodinstallments have to make up for suboptimal preparations in period 1 and 2. This mightbe the considered trade off for the robustness achieved.

For a second example the calculation is done again with different, but still uniform, ran-domly distributed values. The results thereof are shown in figure 4.6 again the bars showthat Stochastic Programming and Worst Case All Periods behave inversely. Here scenariosone, two, or three, four, can be considered together. The first group being slightly more ex-pensive if Stochastic Programming is applied, the latter giving higher cost for Worst CaseAll 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

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 2 3 4

Scenario

Co

st r

elat

ive

to A

ll P

erio

ds

cost

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 onefor three and four.

28 CHAPTER 4. RESULTS

4.2 Uncertainty in Demand Load

DEM(1)

DEM(1)

DEM(2,{1,2})=0,3*DEM(1)

DEM(2,{1,2})=0,4*DEM(1)

DEM(2,{3,4})=0,7*DEM(1)

DEM(2,{3,4})=0,6*DEM(1)

DEM(3,1)=0,3*DEM(2,{1,2})

DEM(3,1)=0,4*DEM(2,{1,2})

DEM(3,2)=0,7*DEM(2,{1,2})

DEM(3,2)=0,6*DEM(2,{1,2})

DEM(3,3)=0,3*DEM(2,{3,4})

DEM(3,3)=0,4*DEM(2,{3,4})

DEM(3,4)=0,7*DEM(2,{3,4})

DEM(3,4)=0,6*DEM(2,{3,4})

Example 1

Example 2

Figure 4.7: Scenarios for uncertainty in demand increase

Another uncertain parameter considered is the development of the demand load. This isa realistic case, as observations have shown the number of demands may rise by hardlypredictable rates. If with a simple All Periods model the demand is not predicted correctlypotential savings from smart dimensioning the network in early stages are wasted. Us-ing the Stochastic Programming planning method, several scenarios of periodical demandincrease are included in planning.

In Figure 4.7 the demand scenario tree is shown. For the first PEriod, the demand is aknown uniform random distribution. The following periods the demand allocation is thesame. The demand matrix is multiplied by a factor ≤ 1 the following period and added tothe 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 previoussection (Section 4.1). The word ‘mean’ in this context means the alternatives are 0.3 and0.7 (0.3+0.7

2= 0.5) in the first example and 0.4 and 0.6 in the second one (0.4+0.6

2= 0.5).

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 distributedover the network . This provides feasibility for all approaches and a reasonable max nodaldegree 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 issmaller in this case, assuming known allocation and uncertain increase, than in the case ofpartially unknown allocation. The performance of incremental is better with an expectedvalue 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 withina one percent margin, this time SP is expected to cost 4% more than AP. Still a gain ispossible 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

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

All Periods StochasticProgramming

Realistic - All Periods Worst Case - AllPeriods

Incremenal

Approach

Exp

ecte

d c

ost

rel

ativ

e to

All

Per

iod

s co

st

Figure 4.8: Expected costs for Uncertain Demand Increase At Known Locations. Example1

Expected cost for uncertainty in demand increase. Example 2

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

All Periods StochasticProgramming

Realistic - All Periods Worst Case - AllPeriods

Incremenal

Approach

Exp

ecte

d c

ost

rel

ativ

e to

All

Per

iod

s co

st

Figure 4.9: Expected costs for Uncertain Demand Increase At Known Locations. Example2

30 CHAPTER 4. RESULTS

Comparing the Stochastic Programming and Worst Case - AP/ Real - AP approachesthe high cost of SP in Figure 4.8 is obvious. This is due to the high variance betweenscenarios. 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 takesplace in case for WC-AP as well as SP comparing the plan for scenario 4 with a realizationof scenario 1.

It is important to recall the definitions of Real - AP and Worst Case - AP given at thebeginning of this chapter. Real - AP seems fairly better than the Worst Case - AP inFigure 4.8. Still then equal probabilities for all scenarios and equal probabilities to planfor the right scenario are rather unrealistic. The area between the two values gives anestimate of a realistically achievable solution.

If the differences between scenarios are not as big the performance of SP significantlyimproves and costs are expected to be below AP planning with a possibly wrong assumptionof 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 thehigher increase towards the end of forecast horizon, namely scenarios two and four, resultin 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 notresult in the maximum.

Figure 4.12 lists the cost for installed equipment for a three period plan and various ap-proaches. It provides an overview of how the total cost, which to minimize is the aim ofthe optimization, is composed. Value shown are absolute costs in cost units, accordingto the NOBEL2 cost model. The category axis is divided in Approach and then furtherdown to periods. This allows a comparison of costs for individual periods. The first stagecosts are about the same for the multi period approaches. Incremental planning howeverhas significantly lower first period costs, ten percent well below the others. This suitsthe pay-as-you-grow investment approach: first stage Capex are as low as possible. If nogrowth takes place money is saved. If growth occurs expansion is more costly but this canbe compensated by increased revenue, caused by the growth.

4.2. UNCERTAINTY IN DEMAND LOAD 31

Absolute cost for uncertainty in demand increase. Example 1

0,00

1000,00

2000,00

3000,00

4000,00

5000,00

6000,00

7000,00

1 2 3 4

Scenario

Co

st u

nit

s

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

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 2 3 4

Scenario

Co

st r

elat

ive

to A

ll P

erio

ds

cost

All Periods Stochastic Programming Worst Case - All Periods Incremental

Figure 4.11: Costs For Uncertain Demand Increase At Known Locations

32 CHAPTER 4. RESULTS

Detailed

equ

ipm

ent co

st for o

ne scen

ario

0

500

1000

1500

2000

2500

3000

3500

4000

12

31

23

12

31

23

All P

eriodsS

tochastic Program

ming

All P

eriods - Worst C

aseIncrem

ental

Perio

d;

Ap

pro

ach

Cost units

Transponder

Regenerator

OM

S

Node

Figu

re4.12:

Com

position

oftotal

costfor

various

approach

es

4.3. VARIATIONS IN COST DECREASE 33

4.3 Variations in Cost Decrease

cost(1)

cost(1)

cost(2,{1,2})=(1−0,1)*cost(1)

cost(2,{1,2})=(1−0,1)*cost(1)

cost(2,{3,4})=(1−0,3)*cost(1)

cost(2,{3,4})=(1−(−0,1))*cost(1)

cost(3,1)=(1−0,1)*cost(2,{1,2})

cost(3,1)=(1−0,0)*cost(2,{1,2})

cost(3,1)=(1−0,3)*cost(2,{1,2})

cost(3,1)=(1−0,3)*cost(2,{1,2})

cost(3,1)=(1−0,1)*cost(2,{3,4})

cost(3,1)=(1−(−0,1))*cost(2,{3,4})

cost(3,1)=(1−0,3)*cost(2,{3,4})

cost(3,1)=(1−0,2)*cost(2,{3,4})

Example 1

Example 2

Figure 4.13: Scenarios for uncertainty in cost decrease

Up to this point uncertainty was assumed to influence the demand. However obviouslythere are other factors of uncertainty in network planning. One of the additional uncer-tainties is uncertainty in the development of equipment prices. Emerging new technologiesare lowering prices, however actual values and dates are hard to predict, as developersusually keep inventions to their selves until the market launch. Additional uncertainty isdue to economical circumstances, benefits from special offers and economy of scale savingsare possible, but it is also possible, the global economic situation causes prices to stag-nate instead of fall. An act of nature beyond control may cause temporary or permanenttrouble, 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. Thusit is interesting to compare the robustness against demand fluctuations with the robustnessconcerning variations in equipment prices.

To model realistic cases, scenarios (see Figure 4.13) are chosen to include steep, gentle, orno decline as well as unpredictable changes together with sudden increase of equipmentcost. The demand is assumed to be known completely to measure only the influences ofthe cost decreasing factor.

In Figures 4.15 and 4.14 the results are shown. To maintain the scale used in all othercharts, in Figure 4.14 the cost-bars for incremental are extended in dashed lines. It becomesobvious, that a very gentle cost decrease, or worse a cost increase, is a disadvantage forIncremental planning. Generally the true multiperiod approaches SP and both(R, WC)-APare very close to the optimal solution, meaning within a one-percent gap. In Figure 4.14scenario three shows a peak in WC - AP. In scenario 3 we assume that equipment costcontinuously rises, therefore an ill-conceived investment plan relying on savings in lateperiods fails badly.

34 CHAPTER 4. RESULTS

Detailed cost for uncertainty in cost development. Example 2

1,15 1,13

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 2 3 4

Scenario

Co

st r

elat

ive

to A

ll P

erio

ds

cost

All Periods Stochastic Programming Worst Case - All Periods Incremental

Figure 4.14: Detailed cost for uncertainty in cost development. Example 2. Dashed barsin scen. 3 and 4 excessed common upper limit of thesis’ standard chart

Limitations of regarding cost uncertainty with the mentioned models apply due to con-straint implying no channels be installed prior to their respective use. Other than thatoperational expenditure (opex) would have to be considered in addition to capital expen-diture (Capex). This however is not the scope of this thesis.

The probabilities of scenarios should be chosen with respect to economic and market situ-ations. Risk of large expenditures late in the lifetime can be lowered by including possiblyrising costs.

In [MSE08] it is stated that the incremental approach performs better for a higher costdecreasing factor. Remembering the basic principle of the approach this is quite obvious: asit is optimizing each period on its own the INC approach delays purchasing of equipmentas long as possible. If the equipment constantly gets cheaper the effect of suboptimalrouting in later periods and the associated need for longer fibers, higher nodal degreesetc. is compensated by the equipment being cheaper in late stages. Naturally the effect isreversed 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

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

All Periods StochasticProgramming

Realistic - All Periods Worst Case - AllPeriods

Incremental

Approach

Exp

ecte

d c

ost

rel

ativ

e to

All

Per

iod

s co

st

Figure 4.15: Expected cost for uncertainty in cost development. Example 1

36 CHAPTER 4. RESULTS

4.4 Preventing Infeasibilities in Routing

DEM(1)=rand(0..20)

DEM(2,{1,2})=rand(0..10)

DEM(2,{3,4})=rand(0..10)

DEM(3,1)=rand(0..10)

DEM(3,2)=rand(0..10)

DEM(3,3)=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 networkcan become quite loaded. For those highly loaded networks planning routing ahead of timeis especially important. If routes are chosen only with current demand and minimizationof current costs in mind, blocking may occur quickly. In our model we limit the nodaldegree to ten, and allow only one node at specific nodal site. With respect to the usedUS-Network-Topology (4.2) the PA node turns out to be heavily used, lying on one of theshortest paths for a lot of node pairs. If nodal degree ten is reached, no further demandscan be routed if there is no capacity for OChs left in the already present OMS. Thissituation is not only reached, if there are a lot of demands originating or terminating atthe bottleneck node, PA but also if there are a lot of demands between nodes adjacent toit. In this case the node has to be traversed by OChs between other node pairs, using upall 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. Thisresults 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 distributionsare of equal mean demand size, but differ in individual demands between node pairs. Thusthe uncertainty is present, comparable to the Allocation of Demand Uncertainty studyin Section 4.1. Now however no allocation is assumed to be known, as opposed to thehalf-knowledge assumed in Section 4.1.

Depending on the actual size and distribution of the demand several cases can be distin-guished:

1. No blocking at all: expected high costs for Inc show, performance of Real - AP andSP 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 loadednetworks.

2. Blocking in INC: Incremental approaches work only for a part of the consideredscenarios. Some lead to blocking in late stages as the needed nodal degree exceedsten. Real AP, SP do not lead to infeasibilities and result in costs comparably andefficient, close to the optimum. SP seems to be a little more efficient for those highdemand values. This is due to randomness although minimal, it is present and provesa 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 solutionas usual. Real AP is not able to avoid blocking for all cases. SP with its specialcharacteristic is able to avoid blocking altogether. Costs for SP are close to optimumdue to the large number of installed OCh throughout the network. There simply arenot many alternatives to the given solution.

Further added demand will lead to infeasibilities in SP and at some point not even an APplan will succeed.

Costs / feasibilities for very-high-demand scenarios

BLOCKING

BLOCKING

BLOCKING

BLOCKING

BLOCKING

BLOCKING

BLOCKING

0,00

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 2 3 4

Scenario

Co

st r

elat

ive

to A

ll P

erio

ds

cost

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 shownin Figure 4.17. As suggested blocking occurs for every scenario, using the Incrementalplanning approach. Stochastic programming leads to good results, close to the optimal All

38 CHAPTER 4. RESULTS

Blocking probabilities

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Stochastic Programming Realistic - All Periods Worst Case - All Periods Incremental

Approach

Pro

bab

ility

Routing feasible Blocking occurs

Figure 4.18: Blocking Probabilities

Periods solution. The same effect was observed for Allocation of Demand Uncertainty inSection 4.1. Seemingly this kind of uncertainty does not have a highly deteriorating effecton the costs for planning with Stochastic Programming. Additionally the very high loadof the network does not leave many alternatives for routing, thus leads to results in thesame order of magnitude. The drawback of Worst Case - All Periods is that even thoughone scenario is under all circumstances feasible, it turns out to be rather expensive. Atthis level of saturation of the network very long routes may have to be taken, resulting inhigh costs for unexpected ‘detours’.

In Figure 4.18 again the case of blocking in Real - All Periods is pictured. A probabilityof feasibility of 100% for SP is only achievable for certain if the realizations are limited tothe four scenarios considered. Otherwise of course even with SP blocking may occur.

These results are considered the greatest advantage of Stochastic Programming. Evenif Incremental Planning adds flexibility and prevents overprovisioning, it does limit thenumber of periods with demand increase. In order to resolve the block in a node, reroutinghas 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 onlyvery little certainty about future developments planning ahead can save money. In everycase considered, the costs of Incremental Planning define the upper bound of costs, thosefor All Periods the lower bound. The Stochastic Approach leads to results in between. AllPeriods costs can not be achieved realistically, the possibilities for approaches with muchbetter cost effectiveness are limited though, since the Stochastic Programming results arealready very close to the optimum. Often the choice of model used will be limited by whatdata is available. As long as the uncertainty does not reach a level where no predictionscan be made at all the Stochastic Planning approach can be used to increase robustness oflong term plans. If no single assumption about the future can be made it is questionableif an investment should be made after all, since it resembles a gambling game more, thanit is a sensible economic decision.

As a foundation for planning a network in the real world the model for demand uncertaintyand cost uncertainty should be combined into one to offer the most possibilities to includeuncertainty. Since this thesis only included a Sensitivity Analysis, as a next step Heuristicsmust be developed and evaluated, after that simulations will give statistical properties.Only then the real performance of the Stochastic Programming approach can be expressedin numbers.

Still then decisions must be made wisely, all approaches presented can only result inplanning-tools. Also those approaches have to be chosen according to and possibly beadapted to special environments. Using the wrong tool can result in a faulty decision. Inthe whole decision process, the economic environment plays a major role, cost considera-tions cannot be made without analysis of the market situation. Present competitors andpossible partners for cooperation must be identified, as greenfield solutions are rather rarenowadays.

Appendix A

Abbreviations

CDF Cost Decreasing FactorCapex Capital ExpendituresDCF Dispersion Compensating FiberDGE dynamic gain equalizerAmpl A Mathematical Programming LanguageOLA Optical Line AmplifierLP Linear Programming or Linear ProgramILP (Mixed) Integer Linear Programming or Integer Linear ProgramOMS Optical Multiplex SectionOCh Optical ChannelEOL End Of Life PlanningINC Incremental PlanningAP All Periods PlanningSP Stochastic ProgrammingWC-AP Worst Case All Periods PlanningRO Robust OptimizationRWA Routing and Wavelength AssignmentWA Wavelength AssignmentWDM Wavelength Division Multiplexing

Table A.1: List of Abbreviations

List of Figures

2.1 Illustration of components needed to connect two nodes with one OpticalChannel on one Optical Multiplex Section . . . . . . . . . . . . . . . . . . 6

3.1 Sample scenario tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Illustration of results given by Stochastic Programming model . . . . . . . 20

4.1 Overview of Cost-Dependency on Factor of Uncertainty . . . . . . . . . . . 224.2 US Topology used in models . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Scenario - Location (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Expected Costs - Location (1) . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Scenario Costs - Location (1) . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Scenario Costs - Location (2) . . . . . . . . . . . . . . . . . . . . . . . . . 274.7 Scenarios for uncertainty in demand increase . . . . . . . . . . . . . . . . . 284.8 Expected Costs - Increase (1) . . . . . . . . . . . . . . . . . . . . . . . . . 294.9 Expected Costs - Increase (2) . . . . . . . . . . . . . . . . . . . . . . . . . 294.10 Absolute Scenario Costs - Increase (1) . . . . . . . . . . . . . . . . . . . . 314.11 Scenario Costs - Increase (1) . . . . . . . . . . . . . . . . . . . . . . . . . . 314.12 Composition of total cost for various approaches . . . . . . . . . . . . . . . 324.13 Scenarios for uncertainty in cost decrease . . . . . . . . . . . . . . . . . . . 334.14 Scenario Costs - Cost Decreasing (2) . . . . . . . . . . . . . . . . . . . . . 344.15 Expected Costs - Cost Decreasing (1) . . . . . . . . . . . . . . . . . . . . . 354.16 Scenarios for provoked blocking . . . . . . . . . . . . . . . . . . . . . . . . 364.17 Comparison of approaches’ costs/feasibility for very high demands. . . . . 374.18 Blocking Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

43

List of Tables

2.1 Relative cost values provided by NOBEL 2 multilayer cost model . . . . . 52.2 Different Multiperiod Approaches Used in This Thesis . . . . . . . . . . . . 82.3 Levels of Uncertainty According to Grover [LG05] . . . . . . . . . . . . . . 11

A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

45

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