Steering of Roaming: a Model Design

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Steering of Roaming: a Model Design

J. R. van Workum s1344110

Master Thesis Operations Research Supervisor: dr. B. (Boris) Goldengorin

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Abstract

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1 Introduction

For most people, roaming is a vague section on their telephone bill when been abroad. Therefore, roaming is often concluded to be calling abroad. Actually, the official definition by the GSM-association is not very different. Roaming is defined as the ability for wireless customers to automatically make and receive voice calls, send and receive data, or access other services when traveling outside the geographical coverage area of the home network, by means of using a visited network. Operators usually are both the owner and the user of the network. Subscribers of operators, especially business clients, need to use their phone when going outside the geographical coverage of a network. The coverage is usually restricted by the boarders of a country. In larger countries, like Brazil, Russia and India, coverage is restricted by regions. To enable subscribers to make calls abroad, they are allowed to use the network of an operator in the country visited. To prevent the mix up of the word operator, home operator is used for the operator in the subscribers home country and visited operator for the foreign operator whose network is visited. So, an operator can be both home or visited depending on the subscriber using the network.

The use of another operators network is not limited to voice. Data services are becoming more and more important. Internet, email, television and radio are examples of services provided on a mobile phone. With respect to roaming, all these services combined are called traffic. There are two types of traffic between operators. From an operators point of view, there is traffic to other operators and traffic from other operators. Traffic from a home operator to visited operators is called own traffic and the returned traffic from operators visiting the home operators network is called visitor traffic. Of course, operators charge each other for the use of their network. Operators have to pay for their own traffic and receive money for visitor traffic.

Roaming is expanding and large amounts of money are involved. For operators, roaming is an important part of their revenues. Therefore, oper-ators pay attention to the prices charged for roaming. All operoper-ators have a standard public price scheme for foreign operators using their network. In some cases, it is interesting for an operator to negotiate personal agree-ments (e.g. lower prices) with other operators. Especially, if an operator sends more traffic to, then it receives from a specific visited operator. Then, an operator is on balance outpaying and a reduction in prices leads to a reduction of the total amount paid.

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If lower prices are negotiated, an operator also needs to be able to use these low price operators. To profit from these price negotiations, operators are able to steer their roaming traffic. Tools make sure that certain operators are preferred and other operators are blocked. In this way, traffic can be steered to visited operators. These visited operators can return the favor by steering traffic to the home operator. The steering of traffic is closely related to the existing agreements with visited operators. The interaction of the streams of traffic between operators is of great importance, because steering traffic has influence on the prices and the received traffic, and therefore on the costs, the revenues and the profits.

For example, a simple situation is sketched in Figure 1, where a home operator has to chose a visited operator to use. The network of operator A is bounded to the boarders of the Netherlands. The subscriber of A is in Italy, so the Dutch home operator of the subscriber needs to use the network of one of the Italian visited operator, named X, Y or Z. In this thesis, the goal is to model this decision problem between visited operators. Therefore, a model is designed and analysed for the steering of roaming. In this section the situation is briefly discussed. In section 2 the situation is formulated more precisely and the complicating factors are discussed. The actual model design is given in section 3. In section 4, the practical use of the model is described and discussed. In section 5 numerical examples are analysed. Section 6 summarizes and concludes the model design and the model analysis.

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2 Problem Formulation

The objective of an operator is to maximize profits while ensuring quality. For an operator, own traffic generates costs and visitor traffic generates revenues. If a client of operator A is abroad and uses the network of operator B, operator A will have to pay operator B for using his network. This own traffic of operator A leads to costs for operator A. Vice versa, if a client of operator B uses the network of operator A (visitor traffic of operator A), operator B will have to pay operator A. This will lead to revenue for operator A.

An operator has tools to steer his own traffic. In most cases, the operator determines which foreign operator will be used. Operators try to maximize their profits by steering their own traffic. An easy solution would be to steer as much own traffic as possible to the cheapest visited operator in each country. However, there are complicating factors which are described in the next sections.

2.1 Correlation in Traffic

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2.2 Pricing Agreements

For the use of another operators network, a price is paid. Each operator has a standard public price scheme for the use of his network. However, operators are able to make price agreements with other operators. In price agreements also the volumes of traffic or a relation between volumes and prices can be included. Most agreements are bilateral and cut both ways, in the costs (own traffic) and in the revenues (visitor traffic). Lower prices for own traffic reduce the costs. Because most agreements are bilateral, the price for visitor traffic is also lower. However, this does not have to cause a decrease in revenues. A lower price makes it attractive for the other operator to use the home operators network. So, if the volume of visitor traffic increases enough, it will compensate the lower price and cause an increase in revenues. A loss in revenues can also be compensated by a decrease in costs. Most agreements contribute to the profits of both operators. Operators are indifferent about the way this is realized; a cost and revenue reduction, a revenue and cost increase or a combination of both. Therefore, the price is often related to the volumes of received traffic.

There are three common used agreements. Operators can vary on these standards.

• Send-or-pay; a predetermined amount of traffic, called a baseline, is committed for a fixed amount. This fixed amount is paid even if the baseline is not met. If there is extra traffic, the actual amount send is more then the predetermined commitment, then another price is charged for this extra traffic, but not for the commitment.

• Block-pricing; if predetermined baselines are met, a decrease in the price is agreed. In agreements multiple baselines can be included. So for zero traffic till the first baseline a high price per unit of traffic is charged. In between the first and second baseline a lower price per unit of traffic is charged and so on. For each amount of traffic between baselines, called blocks, a different price is charged.

• Growth-pricing; this agreement is very similar to block-pricing. Also predetermined baselines with price decreases are included, only the price decrease is for each unit of traffic. So if a baseline is met, the same lower price is charged for all minutes. With block-pricing all blocks have a different price, where with growth-pricing all minutes have the same price.

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The knowledge about agreements between other competitive operators is minimal. They only know their own prices, their amount of traffic and the amount of traffic of other operators on their network.

2.3 Group to Group Agreements

So far only individual operators are discussed. However, operators are able to group themselves. For instance, if one company owns operators in several countries, then all these operators together can be seen as a group. The operators within this group charge each other favorable price. If an operator outside the group makes an agreement with a group, the agreement is applied to each individual operator in this group. Of course, two groups can make an agreement as well. Then, each individual operator in one group has an agreement with all operators in the other group. Groups also have the possibility to exclude one of their individual operators from an agreement.

The group negotiations are often seen as hard, because the interests of the individual operators in a group differ. Group agreements can result in extra profits for the group, but in a negative result for an individual operator. An exclusion of this operator might be profitable for one group, but leads to a worse result for the negotiation partner. Further negotiations or compensation is needed to agree terms.

All the possible varieties in an agreement make it often hard to keep an overview. For example, include or exclude individual operators, baselines for the group or individual operators, one price for the group or different prices for each operator. In group-to-group agreements, these varieties can be included on both sides of the agreement.

After agreements are signed, steering is used to satisfy the agreed con-ditions. Group-to-group agreements complicate the choice of operator, be-cause there is overlap of the operators in the different countries. The cover-age of networks is often restricted by the boarders of a country, so steering is possible within countries. Groups are active in several countries, so a choice need to be made in which country to steer the traffic to which operator, such that profits are maximized and all agreements are met.

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Figure 2: A schematic representation of the complicated situation in the Nordics 2.4 Quality of Service

Steering is possible to a certain point. In some remote regions an operator may not have full coverage, so steering to this operator is not possible. Also, some operators do not offer all services or have the required network proper-ties. This reduces the choice of visited operators as well. So, steering in this case is limited. Operators with a larger coverage area or quality (network properties and services) are sometimes guaranteed of traffic. Subscribers can also divert from their operators steering by manually choosing the visited operator.

2.5 Research Question

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2.6 Purpose of Model

The market is becoming more and more complicated. All operators are aware of the importance of visitor traffic. In the past the focus was mainly on costs. So, the objective was to buy traffic at the lowest rates. The awareness of the balancing effect of visitor traffic is growing. If one minute is sent to and received from an operator at the same price, the own minute is on balance send for free.

If there are no groups of operators active, the problem can be divided in small problems for each separate country. However, more and more op-erators are operating in groups to strengthen their negotiation position. Therefore, contract negotiations become more difficult. An operator negoti-ating with a group, needs to have an overview of the activity, traffic streams and contracts in all countries where this group is active. If in one of these countries another group is active, then all the countries where this group is active need to be included as well. A decision can no longer be made on country level. The impact of a decision does have consequences in related countries and for related operators.

This thesis is written for academic purposes. A problem encountered in real life is modeled with mathematical techniques. The translation from a non defined problem enclosed in an operational situation to a well-defined mathematical model is the academic goal of this thesis. The solutions and post optimality analysis are included to show the possibilities and function-ality of the model. The practical value is however of less importance. The purpose is not to make the model operational, because calculation times should be very small.

The applications of the model indicate how the model could be used to increase the understanding of the market. The relations of a home operator or group with the visited operators and groups are examined. Insights are provided in problems, where the solution is not straightforward. The model gives a steering proposal; how much traffic should be send to each individual visited operator to maximise profits. The important part that this steering proposal encounters is the combination of different aspects. The amount of visitor traffic, the quality of the operator, the prices agreed in a contract or a standard price scheme and relations between operators in different countries are all taken into account to determine the optimal steering proposal.

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2.7 Literature Review

This section gives a review on the literature related to the problem in this thesis. The objective of this thesis is to design an optimization model for the streams of traffic, both own and visited, such that profits are maximized, taking into account the restricting elements, such as pricing agreements, traffic correlation, quality of service and group to group agreements. To the best of my knowledge, there is no literature combining all these restric-tive elements. Although, there is literature discussing similar situations in telecommunication or procurement dealing with one or several of the restric-tions.

For the interested reader, Sutherland (2001) is an article with some back-ground reading on roaming. In this article some technical details are ex-plained. Also, the price regulation from the European Union on the roam-ing charges is discussed. The roamroam-ing situation is a problem faced with by telecommunication operators. Most literature on telecommunication is however on network design or pricing problems. The articles about network design have a more technical nature and are not applicable in this situation. Steering is an operational problem. The vast majority of the operational articles is however about the pricing in networks (e.g. Bouhtou, van Hoesel, van der Kraaij, and Lutton (2007), van der Kraaij (2004)). In these articles, the goal is to set the prices such that revenue is as large as possible. This is not the goal of this thesis. The prices are assumed to be known and the traffic is used to optimize profits, where Bouhtou et al. (2007) and van der Kraaij (2004) use the prices to maximize revenues. However, some parts of these articles are useful.

To get some understanding about how telecommunication problems are dealt with in literature, van der Kraaij (2004) was useful. In this article a tariff or price setting problem of an operator is discussed. The problem is to set prices for using parts of the operators network. The problem is a two-stage game. Firstly, the operator sets its prices and secondly the client chooses the cheapest way to distribute his call. This is modeled as a bilevel program. One of the main assumptions is full knowledge of the market. Especially, the prices of the other operators and the specific demands of the clients are known. The nature and assumptions of this problem do not fit the problem faced with in this thesis. However, the pricing strategies used in the article, are similar to the pricing strategies in this thesis.

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chosen by the home operator for its foreign minutes. These minutes can be seen as ”supply” and the other operators as ”suppliers”, so this is a problem of supplier selection. In all articles, Mixed Integer Linear Pro-gramming (MILP) is used to model their specific situation. In one of the articles, van de Klundert et al. (2005), the supplier selection is focused on the telecommunication market.

In van de Klundert et al. (2005), a problem related to roaming is dis-cussed. In this article, the calls with the start in the home network and the destination in another network need to be routed via carriers such that the costs are minimized. It sounds familiar to roaming, but this is known as interconnection. The problem of roaming is about selecting operators using a network. How these operators make their connections is not of interest to the home operator, however the corresponding quality of the connection is. In the article only one way traffic is concerned, where in this thesis an important aspect is the two way traffic between operators. However, the nature of the problems are quite similar. In both, the article and this thesis, the prices are assumed to be known and the supply, call minutes, need to be divided over the carriers or respectively the operators. For each destination, there are different active carriers and operators. An important aspect is the volume discounts, which are useful in this thesis. The model needed in this case is a further extension of the model introduced in van de Klundert et al. (2005). The group agreements and the relation between own and visitor traffic need to be implemented.

Another approach is to use a procurement or order model. In this thesis, the home operator has a demand of traffic in a country. This demand needs to be met with the use of other operators networks. The traffic needs to be purchased from other operators. So, an procurement optimizing model could be suitable. In the literature, there are procurement models that deal with quantity or volume discounts, e.g. Sadrian and Soon (1994), Crama, Pascual J., and Torres (2004), Goldengorin, Keane, Kuzmenko, and Tso (2007). These three articles use MILP as a modeling technique as well.

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individual agreement. But, overall there are some similarities between the two problems.

In Crama et al. (2004), a company with multiple plants wants to opti-mize his procurement. Each plant needs specific supplies to make products. The company, so all plants combined, has group discount agreements with suppliers. The separate plants also can have local discount agreements, if there is no group agreement. The problem is about a totally different topic, namely about the purchasing decisions of a multi-plant company, taking into account different alternatives of producing and group or local discount schedules. The latter part about local and group discount schedules is appli-cable to the problem faced with in this thesis. The operators in a group are the ”plants” and the group is the ”company” with a centralized procurement plan.

The approach used by Crama et al. (2004) is in several respects suitable for this thesis. However, the relation between the traffic streams is not in-cluded. In none of the references, an operator or company was a supplier and vendor of the same products. So, in the articles the objectives are to minimize costs or to maximize revenues, while in this thesis the combination and relation between cost and revenues plays an important part. Therefore, the profit needs to be maximized. The implementation of the traffic relation is crucial for the suitability of the model to the problem. This thesis com-bines the restrictive elements, such as pricing agreements, group discounts, quality of services and the traffic relation into one model. Mixed integer lin-ear programming seems the right way to model this problem, because this modeling technique is used in almost all articles.

3 Model Development

The home operators i = 1, . . . , I need to divide their traffic to a destina-tion/country k = 1, . . . , K over the visited operators or groups j = 1, . . . , J active in that country. Operators are only active in one country, but there are groups of operators active in multiple countries. The group is seen as one visited operator j with multiple destinations. The group concept is only of importance in the pricing constraints. Pricing agreements can be made with visitor groups. Reaction or quality of an operator is not on group level, each part of the group has his own reaction and quality.

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are applicable to each home operator separately or to the group of home operators.

3.1 Reaction and Pricing of Visitor Traffic

An important aspect is the amount of visitor traffic received for the sent own traffic. The relation between these two types of traffic needs to be taken into account. Without going into details, steering is done with changing settings in some tools. Minor changes in own traffic will therefore not directly result in a change in visitor traffic. However, if changes become too large, the visited operator will react by changing his settings. These new settings will lead to a different amount of visitor traffic. This amount will again stay constant until another amount of own traffic is sent. The visited operator will once again react by changing its settings. There are some breakpoints where these changes in visitor traffic will occur. In between these break-points, the settings are not changed and the amount of visitor traffic will be constant. The reaction of a visited operator is stepwise constant. The reaction intervals n = 1, . . . , N link the own traffic to the amount of visitor traffic.

Another advantage of the stepwise constant behavior of visitor traffic, is the possibility to transform the amount of visitor traffic from units to euros. Because for a constant amount of traffic the price is known, and the volume in units can be transformed to a volume in euros. The transformation to a relation between own traffic and visitor euros makes pricing constraints for visitor traffic superfluous.

3.2 Traffic and Minutes

The traffic between operators consists mainly of four types of services; the minutes called (MOC), the SMS messages send, the volume of data used (GPRS) and the minutes received (MTC) by the subscriber of a home op-erator on a visited opop-erators network. The most important and largest part of the total traffic is the minutes called. Therefore, most operators use the term minutes instead of traffic. The type of pricing agreements, described in section 2.2, are only applied to the amount of minutes. For the other services, there is only an unconditional price for each service without any commitments or baselines.

Steering is not possible for each separate service, so if 50% of the total minutes is steered to a visited operator also 50% of the total amount of SMS, GPRS and MTC is send to that operator. Each change in minutes has the same proportional effect on the other services. Nonetheless, steering is monitored in minutes. A reaction of a visited operator depends on the minutes sent to that operator.

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for reactions. Therefore, the stream of own traffic in the model is defined in minutes. The other services however will cost money. In the objective the costs of other services are included.

3.3 Notation

Variables are defined in lower case letters and parameters are defined in capitals.

Variables

xijkm = amount of own minutes sent from home operator i to visited operator j in country k in the price interval m

yijm =

 



1, if total own minutes from home operator i to operator j is in price interval m 0, otherwise gjm =   

1, if total own minutes from home group to operator j is in price interval m 0, otherwise zijkn =   

1, if total own minutes from home operator i to operator j in country k is in reaction interval n

0, otherwise

vijk = amount of visitor traffic in euros received from visited operator j in country k to home operator i

Parameters

Dik = amount of own minutes of home operator i in country k

Ajk =

1, if operator j is active in country k 0, otherwise

LRijkn = lower threshold of reaction interval n of own minutes from home operator i to operator j in country k

U Rijkn = upper threshold of reaction interval n of own minutes from home operator i to operator j in country k

V Aijkn = volume of visitor traffic of home operator i from operator j in country k for the reaction interval n in euros

LQjk = minimum percentage of total traffic in country k steered to the visited operator j in country k, due to quality reasons HQjk = maximum percentage of total traffic in country k steered to

the visited operator j in country k, due to quality reasons

Gj =

 



1, if the home operators are grouped with respect to visited operator j

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LOPijm = lower threshold of price interval m of own minutes from home operator i to operator j

U OPijm = upper threshold of price interval m of own minutes from home operator i to operator j

LOP Gjm = lower threshold of price interval m of own minutes from home group to operator j

U OP Gjm = upper threshold of price interval m of own minutes from home group to operator j

P Oijm = price for own minutes of home operator i to operator j in price interval m

P OGjm = price for own minutes of home group to operator j in price interval m

P Sij = price for additional cost per own minute of home operator i to visited operator j

P SGj = price for additional cost per own minute of the home group to visited operator j

F COijm = fixed cost for price interval m of own minutes from home operator i to visited operator j

F COGjm = fixed cost for price interval m of own minutes from the home group to visited operator j

Sets

Kj = {k| Ajk = 1} the set of active countries for a visited operator j IP = {j| Gj = 0} the set of visited operators with an individual

pricing agreement with the home operators

GP = {j| Gj = 1} the set of visited operators with a pricing agreement with the home group

3.4 Objective

The objective is not to minimize costs or to maximize revenues, but a com-bination of both. The objective is to maximize the profit of the home group (z). The profit function has a revenue part R consisting of the received payments for visitor traffic and a cost part C consisting of the payments for own traffic.

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interval.

z = max(R(vijk) − CI(xijkm, yijm) − CG(xijkm, gjm))

R(vijk) = P_ijkvijk

CI(xijkm, yijm) = Pijm((1 − Gj)(P Oijm+ P Sij)Pkxijkm+ F COijmyijm) CG(xijkm, gjm) = Pjm(Gj(P OGjm+ P SGj)Pikxijkm+ F COGjmgjm) 3.5 Constraints

The constraints of the problem are the home operators demand, the pricing agreements, the reaction of visitor traffic and the quality of visited operators. These issues are formulated in the following way:

Demand constraints. A home operator i has an amount or ’demand’ of minutes Dik in a specific country k. The streams of own minutes, xijkm, from a home operator i to all visited operators in a country k, regardless of the price interval m, need to match this demand. These constraints for all home operators i and each country k are

P

jmxijkm = Dik ∀i, k

Individual pricing constraints. The discount constraints are separated in two sets of constraints, the first for pricing own traffic from the individual home operators to the visited operator and the second for pricing own traffic from the home group to the visited operator. With a visited operator, there is an individual agreement or a group agreement. If there is no discount agreement, the standard price scheme is applicable. This standard scheme can be seen as a pricing agreement as well, namely as an expensive uncondi-tional one. The traffic from a home operator i to a visited operator j should be in the appropriate pricing interval and in one pricing interval only. This can be stated with the following constraints for each home operator i with an individual pricing agreement with the visited operator j ∈ IP ,

P

myijm = 1 ∀i, j

P

k∈Kjxijkm ≥ LOPijmyijm ∀i, j ∈ IP, m P

k∈Kjxijkm ≤ U OPijmyijm ∀i, j ∈ IP, m

Group pricing constraints. For these constraints the same holds true as for the individual pricing constraints. Only the minutes of the home group, instead of the minutes of the individual home operators, to a visited operator j ∈ GP should be in the appropriate pricing interval.

P

mgjm = 1 ∀j

P

i,k∈Kjxijkm ≥ LOP Gjmgjm ∀j ∈ GP, m P

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Reaction constraints. The minutes from a home operator i to a visited operator j in a country k is in a reaction interval and in one reaction interval only. Based on the reaction interval, the appropriate visitor traffic vijkfrom a visited operator j in a country k to a home operator i is determined. Reactions are on individual level and not on group level. Between each home operator and each visited operator in each separate country, there is a stream of visitor traffic.

P

nzijkn = 1 ∀i, j, k

P

mxijkm ≥ LRijknzijkn ∀i, j, k ∈ Kj, n P

mxijkm ≤ U Rijknzijkn ∀i, j, k ∈ Kj, n vijk =PnzijknV Aijkn ∀i, j, k ∈ Kj

Quality constraints. A network of a visited operator has a certain quality (coverage, service properties, etc.), which guarantees a visited operator of a minimum amount of traffic. The quality also influences the maximum amount of traffic to a visited operator. It is easier to steer traffic to high quality networks with all properties and national coverage, then to steer traffic to smaller networks with less coverage. For each visited operator j in country k a percentage LQjk of the total traffic Dik from a home operator i to a country k is guaranteed. And only a maximum percentage HQjk can be steered to the visited operator j in country k.

P

mxijkm ≥ LQjkDik ∀i, j, k ∈ Kj P

mxijkm ≤ HQjkDik ∀i, j, k ∈ Kj Nonnegativety and integrality constraints.

xijkm ≥ 0 ∀i, j, k ∈ Kj, m yijm ∈ {0, 1} ∀i, j, m

gjm ∈ {0, 1} ∀j, m

zijkn ∈ {0, 1} ∀i, j, k ∈ Kj, n

vijk ≥ 0 ∀i, j, k ∈ Kj

4 Model in Practical Case

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4.1 Data

Some parameters are available in databases. Some parameters are con-structed with data from databases and some parameters are estimated. Especially, the parameters containing information about a visited opera-tor, such as LRijkn, U Rijkn, V Aijkn, LQjk and HQjk are not known exactly. Therefore, estimations are done. The quality of a visited network is based on the experiences of the home operators. Throughout the years, the home operator has collected enough data and knowledge about the qualities of a network. Although, the qualities are not known with certainty. The data gathered over the years however ensure reasonable estimations. Only little information is available for a new operator, then an estimation is based on experiences with other new operators.

The reaction of a visited operator is not known for sure. Although, a reasonable indication can be given by the home operators. Their knowledge about the behavior, the history, the market position and the relation of a visited operator leads to a reasonable estimation. The reaction is measured in intervals, as discussed in section 3.1. What is the maximum of intervals without getting too complicated ad what is the minimum number before loosing touch with reality? How the reaction intervals are constructed and which choices are made is explained in section 4.3. The goal of the model is of importance to this question as well. The purpose of the model is to get an insight in the streams in the market. How do traffic patterns evolve by changes in prices or visitor traffic. Too much detail could lead to large calculation times and hard interpretation of the output.

The rest of the parameters is known with certainty. For example, the demand of minutes is available in databases and constant over years. The parameters of the pricing constraints are available in the form of contracts. However, some calculations are done to extract the parameters from these contracts. These calculations are discussed in section 4.2.

In section 2.2, four different types of agreements are discussed. The agree-ments are transformed to the parameters of the individual or group pricing constraints. For each individual pricing agreement between a home oper-ator i and a visited operoper-ator j, the parameters LOPijm, U OPijm, P Oijm and F COijm represent respectively the lower bounds, the upper bounds, the price and the fixed costs of the price intervals. Obviously, to formulate correct pricing intervals, they should exist, so LOPijm ≤ U OPijm. There is no overlap in intervals, but all possible values of own minutes have a price, so U OPi,j,m−1= LOPijm and LOPij1 = 0, maxmU OPijm = ∞.

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not matter if one minute or the entire commitment is sent. The charges for the total commitment are paid for anyway. Therefore, the price per minute within the first interval is zero and only a fixed price is paid for the total commitment, so LOPij1 = 0, U OPij1 = V1, P Oij1 = 0 and F COij1 = p1V1. Of course, it is also possible to send more minutes then the commitment. A price p2 is charged for these extra minutes. The price for the commitment stays the same, however the fixed costs for the second interval are not p1V1. The correct price for an amount of minutes x ≥ V1 is p1V1+ p2(x − V1) = (p1− p2)V1+ p2x. So, P Oij2= p2, F COij2 = (p1− p2)V1 with the interval bounds LOPij2= V1, U OPij2 = ∞. In practice, only one price decrease after a commitment is included in a contract. However, it is possible to introduce more price steps. These extra intervals are included in the same way. For example, let V2 be the baseline for a price p3, then the second interval changes to LOPij2 = V1, U OPij2 = V2, the third interval is LOPij3 = V2, U OPij3 = ∞ with P Oij3= p3, F COij3 = (p1− p3)V2.

Growth-pricing. A price pi (i = 1, ..., M ) is charged for all minutes, if the corresponding baseline V1, . . . , VM is not met and the previous baseline is met. For example, an amount of traffic x is in between V2 and V3, then the total costs are p3x. The pricing intervals and prices are straightforward. For all m, LOPijm = Vm−1 with V0 = 0, U OPijm = Vm with VM = ∞, P Oijm = pm and no fixed costs F COijm = 0.

Block-pricing. In principle, this agreement looks similar to growth-pricing, but technically it is more like a send-or-pay agreement. Because, not the same price is charged for all minutes. An agreement is made with prices p1, . . . , pM and baselines V0, V1, . . . , VM with V0= 0 and VM = ∞. A price pi (i = 1, . . . , M ) is charged for a minute between Vi−1 and Vi. For example, take an amount of traffic x between V2 and V3. Then, for the minutes between 0 and V1, p1 is charged. For the minutes between V1 and V2, p2 is charged and p3 is charged for the extra minutes in the third inter-val, x − V2. The costs for x minutes are p1V1+ p2(V2− V1) + p3(x − V2) = (p1−p2)V1+ (p2−p3)V2+ p3x. For all m, the pricing intervals are LOPijm = Vm−1, U OPijm = Vm with prices P Oijm = pm and fixed costs for m ≥ 2, F COijm =Pm_i=2(pi−1− pi)Vi−1 and F COij1= 0.

Unconditional. The price p is the same for all minutes without any traffic relation. The first interval is all possible values of own traffic, LOPij1 = 0 and U OPij1 = ∞. Obviously, P Oij1 = p. There are no fixed costs for this pricing interval, so F COijm = 0. If there is no discount agreement, the standard price scheme is used. This is the same as an unconditional agreement. Only the first interval is of importance with a corresponding standard price. The difference is that this standard price is often much higher then an unconditional discount price.

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F COGjm in the group pricing constraint.

The parameters concerning the stream of minutes are known, but as mentioned in section 3.2 the other services (SMS, GPRS and MTC) also contribute to total costs. The cost of the other services, P Sij, is for all services together and per own minute sent. These costs can be calculated because the percentage of total traffic that is steered to a visited opera-tor is assumed to be the same for all services. So, the percentage of own minutes sent determines the percentage of total volume of other services sent. The current streams of traffic from a home operator i to a visited operator j for each service are known and defined as ˆxM OC_ij , ˆxSM S_ij , ˆxGP RS_ij , ˆ

xM T C_ij . The corresponding prices are pSM S_ij , pGP RS_ij and pM T C_ij . The price of the other services per own minute sent is P Sij = 1/ˆxM OCij (pSM Sij xˆSM Sij + pGP RS_ij xˆGP RS_ij +pM T C_ij xˆM T C_ij ). The cost in a situation with a group agreement are calculated analogous.

The previous steps are done for visitor traffic as well, so the correct volumes in euros V Aijknare calculated, including the cost for other services and the correct price interval from the contract. But before calculating the value of visitor traffic in euros, an estimation is needed for the value of visitor traffic in minutes.

4.3 Reaction

The reaction of the operator is important to the model, but hard to measure. A reaction is based on an action, in the form of a change in own minutes of the home operator. As mentioned before, changes are not made very easily, so not a lot of data is known about reactions. Therefore, reasonable estimations are needed for the minutes of visitor traffic. The model is applied and adjusted to the practical situation at KPN. Employees working with visited operators have the required knowledge to make the estimations. The number of reaction intervals influences the accuracy and the workability. An increase in reaction intervals might increase the accuracy. However, more estimations for smaller intervals are needed. A decrease in reaction intervals reduces the number of estimations to be done, but simplifies the reality.

Working with the model in practice showed that three reaction intervals is enough for the situation at KPN. The small number of intervals increases the comprehensibility. The estimations are done in a reasonable amount of time and the output is detailed enough. Including more intervals increases the calculation times significantly and makes it harder to estimate the visitor volumes, because the intuitive feeling is best with three intervals. Three intervals have a clear interpretation, namely the intervals are interpreted as sending little own traffic, sending much traffic and an interval for all traffic in between.

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re-action and the intervals do not overlap. So, for N = 3 rere-action intervals, LRijk1 = 0, U RijkN = ∞ and for all n, LRijkn= U Rijk,(n−1). Three inter-vals means two breakpoints, namely one for the step from the first interval to the second interval and the other for the step from the second interval to the third interval. These breakpoints indicate the difference between low traffic, medium traffic or high traffic to a visited operator j in a country k, which are expressed in percentages of total traffic in country k. These percentages are defined as b1 and b2. In the perception of KPN, little traf-fic to a visited operator is less then 30% of the total traftraf-fic in country k (b1 = 0.3). And much traffic is more then 60% of total traffic (b2 = 0.6). This leads to the values for the parameters, U Rijk1 = LRijk2 = b1Dik and U Rijk2 = LRijk3= b2Dik.

The estimation for visitor traffic is done in voice minutes. The amount of traffic of the other services corresponding to the intervals are determined proportional to the minutes estimation. The percentual difference between the estimations and the actual minutes is extended to the other services. So, all services increase or decrease with the same percentage. The parameter V Aijkn is the summation of all services priced in euros.

4.4 Software

For solving the model, a tool is developed. The models purpose is to get an insight in market patterns. Therefore, the input should be reduced to the minimum and the output should be made comprehensible to the user. Without getting into details, an Excel-file was used to load in data from large databases and the user fills in the additional data, such as reaction intervals and volumes. This data is processed, as discussed in 4.1, by a program called R, (R Development Core Team (2008)). This program also transforms the problem into matrixform, such that it could be solved by a MILP solver called lp solve, (Berkelaar, Eikland, and Notebaert (2008)). R once again transforms the output of the solver to a clear overview. Articles by Buttrey (2005) and Venables and Smith (2006) are of great help by using this solver and software. With the data collected in one Excel-file, it is easy to make changes in the data and resolve the model. This is especially useful, if the user is uncertain about the behavior of the visited operator. For testing sensitivity to changes in visitor data, the calculation times should be reasonable.

4.5 Solvability

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game is of course to get the maximum result for your operator or group. The model is used to solve multiple scenarios, often including the current scenario, such that they can be compared. Therefore, the calculation times should be reasonable. Multiple scenarios are calculated within a day. In this section, it is however shown that the problem is too hard and has calculation times that are too long to be of practical use. The proposed analysis can however be done, but for academic purposes.

The use of the sets IP , GP and Kj, as defined in section 3.3, reduces the number of variables drastically. For instance, the numerical example solved in a latter section consists of three home operators, 8 countries, 18 visited operators, 3 price intervals and 3 reaction intervals. The reduction of the problem size is from 4824 constraints and 3240 variables (1512 integer) to 1110 constraints and 524 variables (236 integer). The sets are of importance to the problem size, and therefore to the reduction of the calculation times. On a computer with a Intel 1.73 Ghz processor and 1 GB RAM memory, the calculation time for a problem with 1110 constraints and 524 variables of which 236 variables are integer valued is already 21 minutes. The calculation times increase rapidly. A problem only little smaller with 639 constraints and 417 variables (225 integer) is solved in 6 minutes. For academic re-search, 21 minutes is within the boundaries of reasonable calculation times. If implemented in practice, calculation times should however be seconds. A somewhat larger problem with 1615 constraints and 819 variables (435 integer) is solved in 3 hours and 50 minutes. The largest problem instance solved has 1955 constraints and 1037 variables (575 integer). The corre-sponding calculation time was huge, namely 53 hours and 20 minutes. The time to solve a problem is exploding with the problem size.

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4.6 Similar Problem Instances

In the previous section, it was shown that the calculation times of the prob-lem grow exponentially with the probprob-lem size. The probprob-lem is hard to solve. However, some special cases of the problem are similar to problems found in literature.

4.6.1 Buyer Decision Problem

If the discount pricing of the problem is simplified and only one home opera-tor is concerned, the problem can be described as the buyer decision problem (BDP) in Goldengorin et al. (2007). The pricing intervals are merged with the reaction intervals to the pricing bands used in the article. All thresholds of the pricing intervals and the reaction intervals are included. A combina-tion of these thresholds forms the pricing bands for a visited group. So, the reaction needs to be transformed to group level instead of operator level. Also, the agreements that are group agreements for the home operators are separated in individual agreements. The business volumes of visitor traffic combined with the fixed cost for a pricing interval are the fixed cost for a pricing band. The prices per minute are the transportation costs. The merge of reaction and pricing is not straightforward. With some assump-tions combining the separate visitor volumes per visited operator to a group total, it is possible to construct price bands or intervals for each home op-erator and visited group. In Goldengorin et al. (2007), it is assumed that the transportation costs and the fixed costs are decreasing with the pricing bands and only one buyer is concerned. A send-or-pay agreement is there-fore not suitable. With some assumptions and restrictions, a special case of the roaming problem is similar to the buyers decision problem. In the article, the similarity with a capacitated plant location problem (CFLP) is shown. The pricing intervals are ”plants” and only one can be chosen. The fixed cost of an interval and the minute price correspond respectively to the opening costs of a plant and the transportation costs. For more details I re-fer to Goldengorin et al. (2007). Also, a Langragean heuristic is introduced to solve the problem.

4.6.2 Minimum Cost Flow

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A MCF model transport a flow from a source to a sink the cheapest way possible, taking into account maximum and minimum restriction of the arcs. The problem in thesis can, once again with some assumptions, be relaxated to a MCF. As with the BDP, the price intervals are merged with the reaction intervals. For each visited group the interval is fixed, so the prices and the business volumes of visitor traffic are known. First, it is checked that with the fixed intervals a feasible solution exists. In other words, that the sum of the upperthresholds in a country is larger then the demand.

The min cost flow for the roaming problem is constructed in the fol-lowing way. A source with a supply of all traffic of the included countries combined is connected to the home countries. The arcs to the countries have a capacity equal to the demand of the corresponding country. Then, the arcs are directed to the home operators that are active in the specific country. Each home operator is connected to the visited groups, separate from each other, with the prices and bounds corresponding to the fixed interval. Each home operator is connected to his own set of nodes representing the vis-ited groups/operators. These visvis-ited operators on their turn are connected to the countries per home operator where they are active. The countries are connected to the sink through edges with capacity equal to the demand of the home operator to that country. Each suitable setting of intervals is solved this way. With the needed assumption and modifications, the prob-lem can be solved with the help of minimum cost flow probprob-lems. The MCF is a familiar problem and well described in literature. Different algorithms for solving a MCF are known.

5 Numerical Examples

This section shows two applications of the model. The first is a simple situation that is extended step by step, to clarify the different aspects of the model with a numerical example. All aspects are combined in a second real-life example. This application also shows some scenario analyses.

5.1 Traffic

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is no visitor traffic. All these concepts are introduced step by step to show the impact on the solution.

In the first situation, only the prices of voice minutes are taken into account. For home operator A, a minute to X costs 0.20 euro and to operator Y 0.25 euro. Home operator B has different prices, namely 0.40 euro to operator X and 0.30 euro to operator Y. Operator A has 2000 minutes to divide and operator B 1000 minutes. The solution to this steering problem is simple. Operator A sends all minutes to the cheapest operator and operator B does the same. So, the traffic from operator A to operator X is 2000 minutes and from operator B to operator Y is 1000 minutes. The total costs for the home operators combined are e700.

The next step is the introduction of extra services (SMS, GPRS, MTC), see section 3.2. The situation described above stays valid for voice minutes called (MOC). In Table 1, the traffic from the home operators (H - column) is shown.

Table 1: Traffic of the home operators

H MOC (in min) # of SMS Mb of GPRS MTC (in min)

A 2000 1000 500 500

B 1000 200 300 200

In Table 2, the prices between the home operators (H - column) and the visited operators (V - column) are shown. The column P O is the price for a MOC minute. As mentioned before, the steering is done for total traffic and not for the separate services. Therefore, the prices of the other services can be combined. The column (P S) shows this combined price for all services per MOC minute, which is used in the model. The tot - column shows the price for sending one MOC minute to a visited operator (P O + P S). For example, if operator A sends 400 MOC minutes, which is 20 % of all minutes, to operator X, then also 20 % of the other services is sent to operator X. So, operator X also receives 200 SMS-messages, 100 Mb of GPRS and 100 minutes MTC. The total costs of all traffic are e240, see section 4.2. That corresponds to a total price of 0.60 euro per MOC minute. The price for a MOC minutes is however 0.20 euro, so the additional costs for the other services in this example are 0.40 cents. In practice, the additional costs are relatively smaller, because the volume of MOC minutes significantly the largest.

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ser-Table 2: Prices of all services in euros H V SMS GPRS MTC P S P O tot A X 0.20 1.00 0.20 0.40 0.20 0.60 A Y 0.10 0.80 0.20 0.30 0.25 0.55 B X 0.20 0.70 0.10 0.27 0.40 0.67 B Y 0.10 1.00 0.05 0.33 0.30 0.63

vices make operator Y preferable above operator X. Operator B still prefers operator Y. Notice that for operator B the difference between operator X and Y is only 4 cents, while all services, except for GPRS, are more expen-sive. The cheaper GPRS price of operator X almost compensates for the more expensive price of the other services. The costs for other services are definitely not negliable. In the remainder of this numerical example, the combined price for all services is used.

5.2 Reaction and Quality

So far only own traffic and the corresponding costs are considered. In this section, the situation of the previous section is extended with the reaction and the quality of the visited operator. First, the quality of the visited operators is implemented. The maximum percentage of total traffic that can be steered to operator X is 90%. This maximum percentage for operator Y is 70%. So, operator Y is the smaller operator with less coverage and a weaker network. If only two operators are active, the minimum percentage of traffic to one operator is the remainder that can not be sent to the other operator, because there is no alternative. The remaining 30% that can not be sent to operator Y is for sure sent to operator X, and vice versa. Therefore, a third visited operator Z is introduced to create a situation where the remaining minutes also need to be divided amongst more operators. The visited operator Z is more expensive then the other two and charges both operators 0.60 euros per MOC minute, 0.20 euros per SMS message, 2.00 euros per Mb of GPRS and 0.40 euros per MTC minute. This results in a total price ofe1.30 per MOC minute (P O + P S = 0.60 + 0.70) for operator A ande1.32 for operator B. Although, the same prices are charged to both operators, a different total price is calculated. This difference is due to the traffic patterns. Operator B has relatively more of the expensive GPRS traffic.

Operator Z is more expensive then operator X and Y, so the solution and the costs of the previous section stays the same. The qualities of the three operators are shown in Table 3.

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Table 3: The low and high quality of the visited operators in %

V low high

X 10 90

Y 5 70

Z 5 90

traffic is sent to the cheapest. The traffic that can not be sent to the cheapest operator due to quality restrictions is sent to the second cheapest operator. So, operator A sends the maximum 1400 MOC minutes (70 %) to operator Y, a minimum of 100 MOC minutes (5 %) to the expensive operator Z and the remaining 600 MOC minutes (25 %) to operator X. Operator B does the same, 700 MOC minutes to the cheapest operator Y, 50 MOC minutes to operator Z and 250 MOC minutes to operator X. The total costs of both operators are e1874,50. So, the introduced quality restrictions result in additional costs ofe144,50.

5.3 Visitor Traffic

The revenues received for the visitor traffic are just as important as the costs paid for own traffic. A cheap operator does not have to be preferred. An expensive visited operator can be preferred, if the operator returns much more visitor traffic. The stepwise behavior of visitor traffic and the corre-sponding intervals are already discussed in section 4.3. The bounds used are the defaults of 30% and 60% of the total own minutes.

In Table 4, the values of visitor traffic, corresponding to the first interval for each visited operator (V - column) to the home operators A and B, are given in the VA1 - column. Analogously, the VA2 column corresponds to the value of visitor traffic in the second interval and the VA3 - column to the third interval. The values in these three columns are used in the model. VA columns are prices times volumes of all services. The other columns give the underlying volumes of the services for the first interval.

The values of the second and third interval are calculated with the value of the first interval. In practice, the MOC minutes are estimated for each interval. The percentual change of the estimated values of MOC minutes between the intervals is applied to the other services. The reaction interval 1 corresponds to low own traffic (≤ 30% of total traffic), 2 corresponds to medium own traffic (30% - 60% of total traffic) and 3 corresponds to high traffic (≥ 60% of total trafic).

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Table 4: The volumes of visitor traffic

H V MOC SMS Gprs MTC VA1 VA2 VA3

A X 1000 400 100 300 e440 e880 e1760

A Y 500 100 100 200 e255 e510 e1020

A Z 700 300 50 100 e620 e682 e750.20

B X 200 50 20 50 e109 e163.50 e245.25

B Y 100 20 20 50 e53 e106 e212

B Z 100 20 20 20 e112 e123.20 e135.52

a factor 1.5 to operator B. The solution is in this case not as easily seen as before. Table 5 shows the solution with visitor traffic included. The OWN - column shows the own minutes, the % - column shows the own minutes in the percentage of total traffic, thee-column shows the money volume of all traffic corresponding to the own minutes and the VIS - column shows visitor traffic in euros. For example, operator A sends 1200 MOC minutes to operator X. This is 60 % of the total 2000 MOC minutes. From the tot-column in Table 2, it is known that one MOC minute from operator A to operator X costs 0.60 euro, which includes all traffic. So, total costs for the 1200 MOC minutes are e720. The own minutes are in the third reaction interval (≥ 60%), so the visited operator reacts with returning e1760 of visitor traffic (see VA3 - column in Table 4).

Table 5: The optimal solution of the example with visitor traffic

H V OWN % e VIS A X 1200 60% e720.00 e1760.00 A Y 700 35% e385.00 e510.00 A Z 100 5% e130.00 e620.00 B X 300 30% e201.00 e163.50 B Y 650 65% e409.50 e218.00 B Z 50 5% e66.00 e112.00

In comparison with the earlier solution without visitor traffic, operator A is sending more traffic to operator X and less to operator Y. The amount of visitor traffic is decisive. It is profitable to send more traffic to the more expensive operator X.

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and Y is not enough for operator B to send the majority of his traffic to operator X. Although, a little bit more is send to operator X. In the previous solution 25 % of total traffic was sent to operator X and in this solution 30 %. This is to increase the amount of visitor traffic, because 30 % of total traffic secures operator B of the second reaction interval with operator X. The solution in Table 6 has total profits of e1472, which is the balance of total revenuese3383.50 (sum of VIS-column) and total costs e1911.50 (sum ofe-column).

In this section, the pricing agreements are included in the example without visitor traffic, but with quality. The inclusion of both visitor traffic and pricing agreements does not contribute to the clarity of the example.

In all the previous stages, the pricing agreements are all unconditional. In this section, these agreements are changed to send-or-pay, block pricing and growth pricing. Operator A has the following agreements; Operator X: a send-or-pay with a commitment of 600 minutes for 0.20 euro per minute and 0.17 euro for the extra minutes. Operator Y: Growth pricing with a first baseline of 400 minutes with a price of 0.50 euro per minute, a second baseline of 800 minutes with 0.20 euro per minute and a price of 0.15 euro if the second baseline is met. Operator Z: Block pricing with 0.60 euro for the first 500 minutes and a price of 0.10 euro for the extra minutes. Operator B has a send-or-pay agreement with operator X including a commitment of 400 minutes for 0.50 euro per minute and 0.40 euro for extra minutes and a growth pricing agreement with operator Z containing one baseline of 200 minutes with 0.60 euro if the baseline is not met and a price of 0.20 euro if the baseline is met. Operator Y still has the same unconditional price of 0.30 euro for operator B. The agreements are only concerning the MOC minutes. The prices of the other services are set to zero, so they do not play a role in the distribution of traffic.

Table 6: The optimal solution with pricing agreements

H V OWN % e A X 600 30% e120 A Y 1300 65% e195 A Z 100 5% e60 B X 400 40% e200 B Y 50 5% e15 B Z 550 55% e110

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agree-ments. The interpretation of the columns is the same as in Table 5. Operator A is filling the commitment with operator X. These minutes are already paid for and therefore free to send. The remaining minutes cost 0.17 euro to send to operator X, while operator Y charges 0.15 euro after the last baseline. The remaining minutes are enough to meet the baseline, so operator Y is cheaper then operator X. Therefore, it is cheaper to send the extra minutes to operator Y. So, no extra minutes are sent to operator X. A comparison of the total expenses for the remaining traffic volume between operator Y and Z show again that operator Y is cheaper. Operator Z has a cheaper rate of 0.10 euro. However, the first expensive block makes operator Z more expensive for the remaining minutes. So, the minimum is sent to operator Z and the remaining 1300 MOC minutes to operator Y.

The decision for operator B is less complicated. The send-or-pay com-mitment is filled and the price of operator Z after the baseline is lower then the unconditional price of operator Y. So, the minimum is sent to operator Y and the rest of the minutes is sent to operator Z.

In general , the commitments of a send-or-pay agreement are filled. After all these minutes are free to send. However, if a situation occurs where a baseline of another operator is almost met, it could happen that minutes in a commitment are not send. A situation, where the obtaining of a baseline with the free minutes from a commitment saves money, is more profitable. For example, if the baseline for operator B with operator Z is raised to 650 minutes. Then, the minutes from operator B to operator Z in Table 6 cost 550 * e0.60 = e330, where sending 100 minutes more costs 650 * e0.20 = e130. So, a decrease of costs by sending 100 minutes extra. This means that not filling the commitment results in a decrease in costs, although the minutes in a commitment are for free.

5.5 Group Agreements

The last extension is the introduction of groups of operators. Operators group themselves to strengthen their negotiation position. Home operators can group themselves in a home group and visited operators in a visited group. First, a home group is introduced. The contract with operator X is changed to a send-or-pay group agreement for the home operators as a group. They have with a combined commitment of 1500 minutes for 0.30 euro and a price of 0.20 euro for extra minutes. This contract holds for operators A and B and the commitment is for the minutes from operator A and B combined. Operator Z has an unconditional agreement with both home operators for 0.17 euro. The agreements with operator Y stay the same. So, a growth pricing agreement for operator A with a lowest price of 0.15 euro after 800 minutes and an unconditional price of 0.30 euro for operator B.

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Table 7: The optimal solution with group agreements H V OWN % e A X 600 30% e180 A Y 1300 65% e195 A Z 100 5% e17 B X 900 90% e270 B Y 50 5% e15 B Z 50 5% e8.50

agreement. The home operators can decide how to fill the commitment. Their objectives are both the same, namely lowest costs for the group and not for themselves. With this objective, operator B is sending minimum traffic to his cheapest operator Z, just to send the maximum to operator X. So that operator A can send less minutes to operator X and more to his cheapest operator Y. Overall this leads to lower costs because operator Z charges operator B 0.17 euro per minute and operator Y charges operator A 0.15 euro per minute. The total costs are e757.50.

Visited operators can form a group as well. A group is active in multi-ple countries, so far only one country is considered, so an extra country is introduced to form a visited group. The country used so far is One and a new country called Two is introduced. The amount of minutes to One is the same 2000 minutes from operator A and 1000 from operator B. Two is a smaller country with 1200 minutes from operator A and 300 minutes from operator B. Once again, the other services are not included. In country One, operator X1 and Y are active and in country Two operator X2 and Z are active. Operator X is now a group of operators consisting of two operators X1 and X2.

The visited group X has separate send-or-pay agreements with operator A and B, so the home operators are not grouped in this example. Operator A has a commitment of 1500 minutes for 0.30 euro and pays 0.25 euro for extra minutes. The commitment for operator B is 1000 minutes with the same prices. Operator Z still charges 0.17 euro per minute to operator A and 0.20 euro per minute to operator B unconditional of the minutes. Operator Y has the same agreements as before, so an unconditional agreement for 0.30 euro per minute for operator B and a growth pricing agreement with two baselines and a lowest price of 0.15 euro for operator A.

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Table 8: The optimal solution for a situation with a visited group agreement H V C OWN % e A X One 600 30% e180 A Y One 1400 70% e210 A X Two 900 75% e270 A Z Two 300 25% e51 B X One 900 90% e270 B Y One 100 10% e30 B X Two 100 33% e30 B Z Two 200 67% e40

minutes is send to operator Y and the remaining 600 MOC minutes to the operator X. The commitment is completed with 900 minutes to operator X in country Two and the remaining 300 minutes are send to operator Z. Operator B does the same, only his lowest price is with operator Z. This solution results in total costs of e1081. Operator A is responsible for the largest parte711 and operator B for e370.

Table 9: The optimal solution including a group-to-group agreement

H V C OWN % e A X One 600 30% e180 A Y One 1400 70% e210 A X Two 730 61% e219 A Z Two 470 39% e79.90 B X One 900 90% e270 B Y One 100 10% e30 B X Two 270 90% e81 B Z Two 30 10% e6

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Two. Overall, this solution results to total costs ofe1075.90, which is e5.10 less then the costs corresponding to Table 8.

5.6 Sensitivity and Stability Analysis

The prices are fixed, but it would be interesting to know what happens if prices change. Therefore, sensitivity and stability analysis are performed on the prices. First, sensitivity analysis is performed on the price from operator A to operator Y. In the optimal solution, traffic to operator Y is 1400 minutes at a price of 0.15 cent. What happens to the optimal value if this price changes? For analysis, the contract of operator Y is set to unconditional with a price of 0.15 euro. This is done to see the effect of the prices only and omit the baselines. Figure 3 shows the price of operator Y against the optimal value.

Figure 3: Sensitivity analysis: price from A to Y against the optimal value

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to operator Y and no further changes in the solution occur by increasing the price.

Another interesting analysis is to examine the stability of the current solution. For which largest change in price does the optimal solution stays the same? Such changes in prices are called tolerances. There is a lower and upper tolerance corresponding to a decrease, respectively increase, of the price. In Table 10, the tolerance based intervals of all prices are given for which the optimal solution stays the same. This holds if only one of the prices is changed and the other prices stay the same. So, in the sensitivity analysis of the price from operator A to operator Y the solution changed by an increase to 0.17 euro and did not change by a decrease in price. So, the tolerances are −∞ and 0.02 euro and the tolerance based interval for this price is [−∞; 0.17]. The tolerances are calculated for the prices for the extra minutes besides the commitment, because this send-or-pay commitment is sent anyway. In case of operator X, this means that a price of 0.25 is used for the analysis. For the other operators, the prices are all unconditional. Table 10: The tolerance based intervals for the prices in the group-to-group agreement example A → X A → Y A → Z current 0.25 0.15 0.17 interval [0.17; ∞] [−∞; 0.17] [0.15; 0.20] B → X B → Y B → Z current 0.25 0.30 0.20 interval [0.17; ∞] [0.17; ∞] [0.17; ∞]

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Operator A Operator B Figure 4: Combinations of prices with same solution.

For operator B, the solution is the same for all combination with both prices above or equal to 0.17. Because the minimum is already sent to the alternatives of operator B. Extra traffic is obtained by setting the prices lower then 0.17 euro. For operator A, the solution stays the same, if the price for operator Y is lower then operator Z and both prices are lower then or equal to 0.20 euro. Operator Y needs to have a lower price the operator Z. otherwise operator Z becomes preferred instead of operator Y. And an alternative of operator B becomes more preferred if one of the prices for operator A is above 0.20 euro.

5.7 The Southern America Example

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5.7.1 Data

All data is given in Table A.17 to Table A.20 of the Appendix. In this section interesting fragments of these larger data tables are presented. The traffic from the home operators to the different countries is given in Table A.17 of the Appendix. Three services are included, namely MOC, SMS and Gprs. The service MTC is omitted, because no costs for MTC are charged between these countries.

Table 11: The MOC minutes to Southern America

Country KPN Base Eplus

Uruguay 3,044,661 140,402 1,532,423 Peru 2,038,797 50,388 830,409 Argentina 2,864,000 95,552 2,826,318 Mexico 1,144,781 601,066 968,345 Brazil 1,062,466 43,388 409,096 Chili 153,334 12,572 116,997 Ecuador 180,120 16,018 127,882 Bolivia 155,003 8,998 64,852

In Table 11, only the MOC minutes from the home operators to the countries are presented. KPN is the largest of the home operators with 57,6 % of the total MOC minutes, Base is the smallest with 5,2 % of the total MOC minutes and Eplus is accountable for 37,2 % of the total MOC minutes. The important countries for the group are Uruguay, Peru and Argentina with 72 % of total traffic. However, for Base Mexico is by far the most important country with 62 % of their total MOC minutes. This indicates the possible conflicting interests of the individual home operators and the group.

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Telcell TIM Digicell Tigo Uruguay X X X X Peru X X Argentina X X X Mexico X Brazil X Chili X Ecuador X X Bolivia X

Table A.18 shows all visited operators and their agreements with the home operators. Also, the group agreements are included and indicated with ”Group” in the column Home. For each service the price for own and visitor traffic is given. Most prices are according the standard price scheme. If there is a discount agreement on MOC minutes, the abbreviation ’T11’ is a reference to the prices and volumes in Table 12.

Table 12 shows each discount agreement of the home operator or group with a visited operator or group (operator/group column). The type - col-umn indicates the type of agreement; ”send” indicates a send-or-pay agree-ment, ”growth” an agreement with growth pricing, ”block” refers to an agreement with block pricing and ”uncon” is used for an unconditional agreement. The most interesting agreement is the group-to-group agree-ment with the Telcell group, with a commitagree-ment of 4,000,000 minutes for 0.20 euro and a price of 0.17 euro for the extra minutes. The agreement is bilateral, so Telcell group has the same commitment and prices.

In this example, the largest number of price intervals is two. The val-ues in Table 12 are used to construct the price intervals for the model, as described in section 4.2. The price p1 shows the price charged for the first interval with an upperbound of V . If there is no upperbound indicated, then it is ∞. The price p2 corresponds to the second interval, which is the last interval and therefore has an upperbound of ∞. All these columns concern own traffic. All agreements are bilateral with the same prices for own and visitor minutes. For visitor traffic, the column Vvis contains the baseline for visitor traffic. Table 12 shows the prices for MOC minutes. The prices of other services are given in Table A.18.

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Table 12: The discount agreements

Operator/group H type p1 V p2 Vvis

Tigo Group uncon 0.20

Telcell Group send 0.20 4,000,000 0.17 4,000,000

Digicell Group uncon 0.26

MiPhone Group uncon 0.30

VOX Group send 0.30 300,000 0.20 300,000

bMobile KPN growth 0.40 94,000 0.20 75,000

bMobile Base growth 0.40 10,000 0.20 20,000

bMobile Eplus growth 0.40 40,000 0.20 155,000

section 4.3 and not given in Tables A.19 and A.20. The bounds are set on the default values of 30 % and 60 %. In section 5.6.3, scenario analysis is performed on these bounds.

5.7.2 Optimal Solution

The solution for KPN, Base and Eplus is given in respectively Table A.21, Table A.22 and Table A.23. The columns Own and Vis correspond to the own minutes send to and the visitor minutes received from the visited op-erator. A marketshare is the percentage of total traffic in a country send to the visited operator active in that country and is given in the column MS. The change with the current steering is given in the columns dOwn and dVis. A negative value indicates a decrease of minutes between the current situation and the optimal solution, so a shift of minutes from this operator to another operator. Vice versa, a positive value indicates an increase of minutes between the current situation and the optimal solution, so a shift of minutes from another operator to this operator.

The optimal solution results in total revenues of e10,520,117 and to-tal costs of e5,932,377. More important the profits are e4,587,740 for the group. In comparison with the current solution, the optimal solution leads to a cost reduction of e914,388 and a revenue increase of e331,661. Accu-mulated to a profits increase ofe1,246,049.

Steering of Roaming: a Model Design

Steering of Roaming: a Model Design

Contents

1

Introduction

2

Problem Formulation

3

Model Development

4

Model in Practical Case

5

Numerical Examples