Volume discount strategies for the provider selection problem in telecommunications under uncertainty

Abstract

Network providers and bandwidth brokers offer a variety of pricing policies based on differentiated quality-of-service (QoS) levels and volume discount schemes. In this paper, a cost minimization problem under various volume discount policies offered during the bandwidth allocation is formulated and solved via a heuristic algorithm. The proposed heuristic algorithm is based on fuzzy set theory. It has the capability of solving complex bandwidth provider selection and task allocation problems in telecommunications by considering a variety of volume discount policies offered by providers. The efficacy of the algorithm is tested under various scenarios to find the optimal strategies for firms and to explore the suitability of the proposed approach.

Keywords

Telecommunication volume discount optimization heuristic fuzzy QoS

1 Introduction

Accessibility to data has been increasing due mainly to the spread of e-services and wireless communications, and consequently, the way users make decisions has also changed to include transactional data (i.e., facts) [13]. In addition, the importance of data communication services has been increasing in today’s intensely competitive market environment with the development of e-commerce [40]. During the last few decades, the telecommunications industry has become increasingly more complex, with continual changes in the diversity of actors, their strategies, and their business models. Several research studies [4 , 32] have classified the current major actors in the telecom market as network operators (backbone providers), virtual network operators (bandwidth brokers), and customers (end-users or firms). Backbone providers utilize network capacity for the delivery of data services. While the backbone providers own the network, bandwidth brokers do not have. They have to lease capacity from the backbone operators to offer services [51].

Firms acquire the capacity that they need to carry out their daily operations either directly from network operators like AT&T or lease it from various bandwidth brokers such as Giglinx Global [65] that offer bandwidths from multiple providers with competitive bandwidth pricing and premium support. AT&T [61] and CenturyLink [63], a global communications and IT services company in the USA, offer pricing with volume discounts. Moreover, some Canadian telecommunications companies such as Telecom Computer [68], TELUS [69] and Saskatchewan Telecommunications (SaskTel), one of the leading Information and Communications Technology (ICT) provider [67], offer pricing with volume discounts. Consequently, McKinsey & Company, a global management consulting firm [66] and Ben Johnson Associates, an economic research and consulting firm [62], reveal the volume discounted pricing in Telecommunications.

Telecommunication network usage includes both traditional data applications and other applications that provide access to interactive communication such as teleconferencing or Internet telephony [57]. Firms acquire a sufficient amount of resource (bandwidth) from network providers to perform their tasks (applications). Consequently, when selecting telecommunication network providers, firms have to consider both cost and providers’ QoS levels. The QoS requirements for a task include several parameters: bandwidth, loss ratio, maximum delay, and jitter (delay variation) [11]. Some models assume [25] that these parameters are deterministic. Most of the previous research studies in telecommunication have modeled uncertainty via probability distributions or discrete scenarios [3 , 43]. Because of limitations and factors that are not accurately known in the real-world environment, there are a growing curiosity in stochastic modeling [21, 34] and fuzzy mathematical modeling [9, 11] approaches to more closely represent real situations. The fuzzy set theory may offer an alternative approach to coping with the imprecision in the real world [7, 58] because of the unavailability and unreliability of historical data that stochastic models tend to suffer from.

1.1 Problem definition and contributions

In this paper, an optimization problem of a firm while acquiring capacity from several providers offer various pricing, discounts, and QoS levels is studied. The optimization problem is modeled from the firm’s (customer’s) standpoint and assumes firms are rational decision makers. Also, interactions between backbone providers and bandwidth brokers are not investigated. Hence, both actors are called providers, and only interactions between firms and providers are studied. Figure 1 represents the central interactions among actors in the telecommunication market structure [51].

Fig.1

A typical telecommunication market structure portraying the relationships among customers, network providers, and operators.

The maximum tolerable delay and jitter levels required to carry out tasks without interruption are represented as fuzzy numbers to avoid inefficiency attained by using deterministic crisp values. Although some studies in the current literature related to network infrastructure and routing considering fuzzy QoS, to the best of our knowledge no previous study deals with how fuzzy QoS constraints affect provider selection and task allocation policies with a discounting scheme. Another contributions of our study are (i) integrating different capacity discount polices into the proposed optimization model, and (ii) analyzing the effect of these polices on firm’s providers selection and capacity allocation strategy. In this context, a novel mathematical model that allows an exact evaluation of available provider choices is formulated, and an efficient and effective heuristic is developed by exploring the underlying combinatorial structure of the problem. Furthermore, the effects on various output performances of changing the dominating discount bandwidth provider are analyzed. An extensive comparison of the proposed technique with a genetic algorithm (GA) is also carried out to show that proposed Heuristic outperforms the state-of-the-art general-purpose metaheuristic approach. To the best of our knowledge, this insight has not previously been presented in the literature.

The rest of the paper is organized as follows: Section 2 provides a literature review on the use of QoS, pricing, discounting policies and fuzzy QoS applications in telecommunication networks. Section 3 presents the problem definition, the assumptions of our model, and the proposed nonlinear mixed integer model with fuzzy QoS constraints and its equivalent crisp formulation. Hence, a novel fuzzy mathematical programming-based heuristic algorithm is also summarized in Section 3. Then Section 4 presents the computational study to show the competence of the proposed algorithm. Finally, Section 5 provides conclusions and future research directions.

2 Literature review

The literature review is organized into two parts since the problem studied in this paper has a rather broad scope. First, studies about QoS, fuzzy QoS and QoS integration with pricing strategies are reviewed. In the second part, the literature related to discounting schemes in the telecom market is reviewed.

2.1 QoS and fuzzy QoS

Network design and operating policies including QoS issues in telecommunications have been very widely researched [1 , 38]. The other stream of studies in the literature comprises pricing and QoS issues from provider’s standpoint [17 , 23]. Some researchers have even tried to develop and present new metrics to represent QoS [1].

The current literature related to fuzzy QoS parameters in the telecommunications can be divided into three groups. One group considers studies related to routing considering fuzzy delay, jitter, and packet loss. Chen et al. [9] argue that stating the QoS levels with deterministic crisp values is complex and not practical. They also claim that QoS requirements by users can be regarded as fuzzy terms in QoS routing problems since vagueness or fuzziness exists in human judgment. Lorenz et al. [35] claim that QoS constraints are commonly denoted as crisp limits. They show that it is vital to describe the imprecision and the uncertainty of the QoS constraints appropriately.

The other group in the current literature about fuzzy QoS parameters mostly focuses on the selection of providers or services under inaccurate or vague data. Robak et al. [44] introduce fuzzy modeling methodology that may be applied for articulating contracts and service level agreements. Wang [52] states that his approach creates advantages for service providers since QoS characteristics are difficult to assess because of the involvement of consumers’ fuzzy perceptions of QoS.

The last group focuses on the integration of QoS with pricing strategies in telecommunications. In this direction, Hua et al. [22] study the effect of resource allocation and advertising decisions on cloud service providers by differentiated pricing between premium and basic offerings. The article also points out market conditions in which QoS investment may lead to higher revenue. Zhang et al. [60] study a problem that is an integration of service level agreements, pricing and QoS guarantees such as average response time of the service. They analyze a market in which two competitor service providers have to decide pricing and service quality strategies to achieve the competitive advantage, and implementation of two strategies, namely, simultaneous and sequential for service levels and prices investigated. Cruz and Liu [14] set up an equilibrium model for pricing and resource allocation in the Network. Like the previous study, the effect of QoS requirements on the pricing is analyzed under the competitive market structure. In addition, resource allocation issues in the network are addressed as a function of equilibrium price, supply and cost structure.

2.2 Discounting Policies

Pricing schemes are formed in the literature as a function of QoS, the amount of transmission, and the transmission rate. It is assumed that the telecom capacity provider’s pricing is rational at least in terms of QoS and bandwidth offered. For example, every provider charges more for higher QoS, but one provider might sell capacity with a higher QoS level at a lower price compared to another provider. Besides, discount strategies such as quantity [24 , 59] incremental [10, 27] and volume [5 , 54] are used induce not only customer demand but also establish coordination between each actor in the market.

Babic and Peric [5] study multi-product vendor selection problem by integration of volume discount and fuzzy methodologies. They model the problem as a fuzzy multi-objective optimization problem with three objectives: minimization of the total purchase cost, maximization of aggregate quality and maximization of reliability indicators of suppliers. The weights of each objective are determined by Analytical Hierarchy Process (AHP), and as a solution approach well-known weighted additive fuzzy operator is utilized by converting fuzzy objectives into one crisp objective.

Lau et al. [30] investigate the performance of volume discounting schemes to coordinate a supply chain effectively, and game theoretical point of view approach for volume discounts in a retail environment is studied in [29]. Wu and Wu [54] analyze the effects of opaque selling strategy when discounts are used as driving force for demand postponement. In the case of opaque selling, discounts are applied to customers’ advance orders, but the actual delivery of goods are determined by the firm. It is claimed by authors that mentioned strategy leads to a diminishment in both safety stocks and capacity waste. The proposed problem modeled as two-stage stochastic programming model with the objective of maximization of expected profit.

Murthy et al. [37] investigate supplier selection problem under volume-based discounts for bundles. The developed mixed-integers mathematical programming model with minimization objective that includes fixed and variable costs related to supplier selection is solved to optimality under single buyer environment. Different from the previous study, Sawik [45] considers both quantity and business volume discounts simultaneously in the supplier selection problem. Moreover, two different models, namely, single and multi-objective mixed-integer programming are developed in which selection of suppliers carried out not only by price but also the quality of purchased parts and reliability of on-time delivery.

Amid et al. [2], Xia and Wu [55] consider multiple objective problems under all-unit price discounts on total business volume. Nevertheless, proposed models cannot handle multiple discount policies simultaneously. Ebrahim et al. [16] study a multi-objective optimization problem for a single item purchasing problem under different discount types one of them being business volume discount policy. Qin et al. [41] study volume discounts and franchise fees as coordination mechanisms for providers and buyers considering price-sensitive demand. They compare the discounts offered by providers when they work both independently and jointly with buyers.

Previously mentioned research studies related to discounting strategies had made contributions to the supplier selection, supply chain design and configuration and sourcing literature, nonetheless, only a very few research addresses the discounting policies in the telecommunications market. For example, the problem of bundling in the wireless telecommunication by Yang and Ng [56]. The problem can be easily converted to a volume discount model because customers may buy cellular phones at a discounted price if they subscribe to a specific service plan. They also point out the conditions under which the mixed bundle strategy performs better than the individual sale and pure bundle strategies. Another research study on volume discount application in telecom can be found [15]. They study the carrier selection problem by considering volume discounts (CSPV) a procurement optimization problem where providers apply joint business volume discounts. Unlike supply chain models, inventory or order costs do not play a role since capacity or call-minutes are not physical items and cannot be stored. The CSPV deals with the tactical issue of the best carrier to order from. Carriers may use a discount on total business volume to price their services. Researchers also show that the costs for carriers are reduced and the selection process is improved by implementing the best cost routing method, which allows clients to deal with quality issues by adding penalties to carriers’ prices or excluding specific carrier-destination pairs. Kasap et al. [26] offer a solution approach based on GA for the same problem given in [25]. They compare three GA based heuristic solution approaches and show their suitability for resource selection and task allocation problems in Telecommunications without considering volume discounted pricing. Two of the proposed algorithms show less performance compared to [25]. However, the other proposed algorithm obtains reasonable better solutions compared to the heuristic given in [25], but it may require too much iteration with an excessive amount of running time. Since our proposed heuristic is an extended version of the ones in [25] and [49], GA based heuristic solution approaches may not be suitable for our model as well.

In conclusion, although there are extensive studies of volume (quantity) discounts in inventory management and supply chain coordination in the literature, there are only a few studies of the problem in the telecommunications domain. The solutions proposed for the inventory and supply chain management cannot be generalized for the Telecommunications since its environment is different. In case of packet loss, retransmission is required, and it reduces available capacity. Real-time applications such as video conferencing are not allowed task splitting although it is possible to split tasks (orders) in inventory or supply chain management domain. In other words, orders can be fulfilled with multiple shipments or vendors. However, real-life applications in Telecommunications require continuous and uninterrupted assigned capacity that is acquired from one vendor. Moreover, real-time applications have fixed execution times and require allocated capacity right at that moment. Hence, using the volume discount concept in telecommunications is one of the contributions of our research study since a large number of studies related to infrastructure and pricing issues in the telecommunication sector have been published, only a few have investigated the bandwidth acquisition strategies of firms [for example 25, 49 and 50]. Hence, this study presents important contributions to the literature as given below.

This study extends (and builds on) the previous ones [for example 49, 50 and 51] by adding volume discounts, which is a commonly observed feature in real-world situations, but has been mostly ignored in telecom literature due to the level of complexity that it brings to the problem definition. Since this study extends (and builds on) the previous ones, obviously they are similar to each other. However, they also complement each other.

Because of the additional complexity that comes with the inclusion of volume discounts; a new and extended heuristics solution is proposed in this study. A new data set (modified version of previous ones in [25, 49]) is created and used to properly analyze the newly stated problem and the proposed heuristic solution.

Turan et al. (2014) article [49] is focused on the comparison of heuristic methods under different fuzzy to crisp constraint transformation algorithms whereas this study mainly focuses on managerial implications— competition amongst the suppliers under different policies considering with or without volume discount. This study also proposes a new and improved way to represent fuzzy constraints for a more realistic and proper definition of the problem.

3 Proposed mathematical model with fuzzy constraints and solution algorithm

The optimization problem of a firm while acquiring network capacity from several providers that offer resources with various pricing, discounting, and QoS schemes is studied.

All tasks accomplished by a firm are divided into two groups, size-fixed, and time-fixed tasks. All traditional data applications are called size-fixed, while all real-time applications and transactions are classified as time-fixed. Moreover, it is assumed that each task requires particular transmission time, bandwidth, and QoS level. Some individual QoS metrics are defined in networks literature, for example, Cisco [64] includes delay, packet loss probability, jitter, connectivity and download time as its QoS metrics in Service Level Agreements (SLAs). Cisco defines SLAs “as a network performance measurement and diagnostic tool that uses active monitoring, which includes the generation of traffic in a continuous, reliable, and predictable manner” [64]. The emerging technologies, as well as increasing number of users, lead to increasingly complicated delivery of SLAs. Hence, providers usually predefine performance thresholds as a part of SLAs. SLAs that state the threshold amount of delay specifying the amount of time takes for a data unit to travel across the network from source to destination and packet loss representing the data dropped or irrecoverably damaged are very common among internet service providers. Some realistic thresholds values that are used for one-way delay are 90ms for west Europe to the US west, 30ms for the US west to the US east and 40ms within west Europe [64].

Unlike the previous example crisp threshold values, the terms of a particular SLA can be ambiguous or vague as well. For example, customer satisfaction is measured with vague terms such as satisfied or partially satisfied [70]. The best tool to deal with vague terms and data is utilizing fuzzy logic tools. Thus, in the mathematical model, the packet loss rate requirement is modeled explicitly and delay and jitter are formulated as fuzzy Composite QoS [38] as $\tilde{Q} = f \tilde{(δ}, \tilde{σ})$ where $\tilde{Q}$ denotes overall fuzzy QoS of resource and $\tilde{δ}$ , $\tilde{σ}$ represent promised vague delay and jitter values respectively.

The required minimum fuzzy composite QoS ( $\tilde{q}$ ) for seamless completion of tasks can be derived in the same manner: $\tilde{q} = f$ ( $\tilde{η}$ , $\tilde{μ}$ ), where $\tilde{η}$ represents the minimum fuzzy delay requirement and $\tilde{μ}$ denotes the minimum fuzzy jitter requirement for tasks. Both value sets are expected to be set by firms’ decision makers. It is assumed that the packet loss rate remains fixed during the contract period, which implies ignoring the stochastic nature of loss rates.

The all-you-can-send pricing scheme [12] in which the firm pays a fixed price for a fixed amount of bandwidth for a fixed period is considered. It is also assumed that a real-time task incurs an opportunity cost whenever transmission rate decreases. Moreover, the QoS level assured by providers and the minimum QoS level desired to carry out tasks are represented as triangular fuzzy numbers (TFN). The reasons for utilizing triangular possibility distribution to represent the imprecision in the constraints are to enrich the computational efficiency and to smooth data acquisition [8, 19]. However, the decision maker may choose any appropriate possibility distributions based on subjective judgments and/or historical data, such as trapezoidal, bell-shaped, exponential, and hyperbolic. Figure 2 shows arbitrary delay membership functions (μ_Delay) for two different bandwidth providers and a task. The assured delay values for both providers and minimum required delay values for the task are modeled as TFNs. It should be noted that for this representative case, Task 1 can be allocated to the capacity leased from Provider 1. Nevertheless, allocation of Task 1 to Provider 2 solely depends on the minimum acceptable possibility that is defined by the decision maker of the firm.

The objective function of the model contains two cost terms. The first is the capacity acquisition cost which reveals various volume discount schemes of providers with all-you-can-send pricing. The second term is the potential opportunity costs such as the cost of degradation in quality for time-fixed tasks, and the QoS switching cost when the decision maker intentionally reduces the QoS level or shifts to competitor provider.

Fig.2

An illustrative example of fuzzy-membership functions of two bandwidth providers and a task.

3.1 Problem notations and model assumptions

The parameters and decision variables used presented in Tables 1–3 throughout the text when setting up formulations and solution procedures. The proposed fuzzy constrained non-linear mixed integer mathematical programming model is based on the following assumptions:

Due dates for all tasks are always later than the ending of the contract period.

A task cannot be distributed among different resources.

At least one resource exists in the market that meets the fuzzy QoS constraint of every task.

An all-you-can-send pricing is applied by all providers at every bandwidth discount level.

The transmission rate of the size-fixed task may drop to zero during execution.

Table 1
The sets used in problem formulation

I The ordered index set of resources

J The ordered index set of tasks

A_T, A_S The index set of tasks with fixed transmission time and fixed size respectively where A_T∩ A_S = ∅ and A_T ∪ A_S = J

Table 2

The parameters used in problem formulation

m _i	Number of bandwidth levels provided by resource i
α _ik	Transmission efficiency, calculated as one minus the packet loss rate of resource i at bandwidth level k
β _ik	Amount of bandwidth that can be leased from resource i at bandwidth level k
c _ik	The unit cost of leasing bandwidth from resource i at bandwidth level k, note that c_il ≤ c_ip holds for ∀p ≤ l where l, p = 1, . . . m_i
L _i	The duration of contract with resource i
C _i	The amount of capacity that can be leased from resource i
$c_{j}^{o}$	The opportunity cost of missing the target transmission rate for task j.
${\bar{w}}_{j}$	The size of task (j ∈ A_s) in number of bits
${\bar{r}}_{j}^{U}, {\bar{r}}_{j}^{L}$	Target and the minimum transmission rate of task j, respectively
$Δ t_{j}$	Estimated scheduled transmission time for time-fixed task, j ∈ A_T
${\bar{Q}}_{i}$	Fuzzy triangular composite QoS level of resource i
${\tilde{q}}_{j}$	Fuzzy triangular composite min required QoS level for completion of task j

Table 3

Decision variables used in the problem formulation

x _ik	1 if bandwidth level k from resource i is selected to resource i, 0 otherwise
y _ij	1 if task j is assigned to resource i, 0 otherwise
y _ijt	1 if task j is active (transmitting) at time t on resource i, 0 otherwise
r _j	The transmission rate of task j.
t _j	Start time of task j.

The first assumption helps us allocate tasks to any place in any resource as long as the resource satisfies the fuzzy QoS requirements. The second assumption arises because of the infrastructural restrictions of telecommunication networks. The third assumption is needed to ensure assignment of all tasks. The fourth assumption implies that an all-you-can-send strategy is chosen by all bandwidth providers, although they may adopt different discounting schemes. The fifth assumption is needed for computational tractability of the model, even though a zero-transmission rate may not be allowed in real-life applications.

3.2 Problem formulation

Telecommunication backbone providers offer different discount schemes to encourage firms to acquire more bandwidth. Further, providers lease several different amounts of capacity (β_ikL_i) with a fixed price for a particular duration and bandwidth. To illustrate, Fig. 3 depicts an allocation scheme of 3 time-fixed (j₁, j₂, j₃) and 2 size-fixed (j₄, j₅) tasks into the provider i. The provider i offers 3 bandwidth levels (m_i = 3), and firms decide to lease capacity at second bandwidth level (β _i2 ) with total capacity C_i = L_i x β _i2 .

Fig.3

A visual representation of the relationships among the problem parameters.

The objective function (1) has two expenditure terms as leasing and opportunity, and deliberates the tradeoff between the cost of acquisition by considering volume price discounts and the cost of not satisfying aimed transmission rates in real-time tasks.

$Minimize \underset{capacity leasing cost}{\underset{︸}{\sum_{i \in I} \sum_{k = 1}^{m_{i}} C_{ik} x_{ik} β_{ik}}} + \underset{opportunity cost}{\underset{︸}{\sum_{i \in I} \sum_{j \in A_{T}} \sum_{k = 1}^{m_{i}} ({\bar{r}}_{j}^{U} - r_{j} α_{ik}) C_{j}^{o} y_{ij}}}$ (1)

Constraint set (2) states that the total size of all tasks to be carried out is guaranteed by the available capacity considering the packet loss rate of resources so that it is utilized up to the adequate usable capacity. For example, total area of tasks both time- and size-fixed in Fig. 3 cannot be larger than the area of rectangle corresponding to dimensions of L_i and β_i2. $\sum_{j \in As} {{\bar{w}}_{j} y}_{ij} + \sum_{j \in AT} α_{ik} Δ t_{j} r_{j} y_{ij}$ $\leq (\sum_{k = 1}^{m_{i}} α_{ik} x_{ik} β_{ik}) L_{i} \forall i \in I$ (2)

Fuzzy constraint set (3) guarantees that a task can be allocated only to resources that offer higher or equal composite QoS levels, where composite QoS levels are triangular fuzzy numbers (TFN) and functions of delay (σ) and jitter (δ). $y_{ij} {\tilde{q}}_{j} \leq {\tilde{Q}}_{i} \forall_{i} \in I \forall_{j} \in J$ (3)

Constraint (4) guarantees that all time-fixed tasks assigned to a resource are completed within the length of the contract duration. A similar constraint is not mandatory for size-fixed tasks because constraint set (2) ensures that a size-fixed task is only assigned to a leased resource if there is enough capacity and it can be completed on time since the transmission rate can fluctuate. That is, size-fixed tasks are not required to be in a rectangular shape. To illustrate, transmission rates of tasks j₄ and j₅ alters during contract period as depicted in Fig. 3. $(t_{j} + Δ t_{j}) y_{ij} \leq L_{i} \forall_{j} A_{T} \forall_{i} \in I$ (4)

Constraint set (5) avoids consuming more bandwidth than the acquired amount at any bandwidth level and any time (bandwidth dimension) during the contract period. To illustrate, the total bandwidth level used at anytime during contract period cannot be over the leased bandwidth level β _i2 in Fig. 3.

That is, y dimension is limited by of $\sum_{j \in J} r_{j} y_{ijt} - (\sum_{k = 1}^{m_{i}} x_{ik} α_{ik} β_{ik})$ $\leq 0 \forall i \in I t = 1, . . ., L_{i}$ (5)

Constraint (6) along with (17) guarantees that a task is assigned to only one resource and all tasks are allocated. Constraint sets (7) and (8) ensure that the time-fixed tasks are allocated to the required number of time slices, and the size-fixed tasks are placed into sufficient capacity, respectively. $\sum_{i \in I} y_{ij} = 1 \forall_{j} \in J$ (6) $\sum_{i \in I} \sum_{t \leq L_{i}} y_{ijt} = Δ t_{j} \forall_{j} \in A_{T}$ (7) $\sum_{i \in I} \sum_{t \leq L_{i}} r_{j} y_{ijt} = {\bar{w}}_{j} \forall_{j} \in A_{s}$ (8)

Constraint set (9) ensures that at most only one bandwidth level from any resource can be chosen. Constraint set (10) ensures that tasks are only allocated to the chosen bandwidth levels of selected resources. In the same manner, constraint set (11) guarantees that a task is assigned to a network resource only if it occupies a time slice on it. $\sum_{k = 1}^{m_{i}} x_{ik} \leq 1 \forall_{i} \in I$ (9) $y_{ij} - \sum_{k = 1}^{mi} x_{ik} \leq 0 \forall \in I \forall_{j} \in J$ (10) $y_{ij} \leq y_{ijt} \forall i \in J \forall \in Jt \leq L_{i}$ (11)

Constraint set (12) states that the transmission rate for a time-fixed task j should be high enough to satisfy the minimum transmission rate at the sink node. Constraint set (13) enforces the upper bound transmission rate limitation. $r_{j} α_{ik} \geq y_{ij} {\bar{r}}_{j}^{L} \forall j \in \forall \in_{T} \forall i \in Ik = 1, . . . m_{i}$ (12) $r_{i} α_{ik} y_{ij} \leq y_{ij} {\bar{r}}_{j}^{U} \forall j \in \forall \in_{T} \forall i \in Ik = 1, . . . m_{i}$ (13)

Constraint set (14) puts a limitation on leasable capacity from each telecommunication resource. $(\sum_{k = 1}^{m_{i}} x_{ik} β_{ik}) L_{i} \leq C_{i} \forall_{i} \in I$ (14)

The remaining constraints are for the non-negativity (15) and integrality (16, 17) of the decision variables, respectively. $r_{j}, t_{j} \geq 0 \forall j \in J$ (15) $x_{ik}, y_{jt} \in {0, 1} \forall j \in I k = 1, . . ., m_{i}$ (16) $y_{ij}, y_{ijt} \in {0, 1} \forall j \in J \forall i \in It \leq L_{i}$ (17)

3.3 Formulation with crisp QoS constraints

The first stage of solving the model proposed in Section 3.2 is to transform fuzzy constraints into equivalent deterministic forms. Fuzzy information and data have usually been described by fuzzy numbers. If there exists a_L, $a_{M}, a_{U} \in R$ such that A (x) = (x–a_L)/(a_M–a_L) for each x ∈ [a_L, a_M) , A (a_M) = 1, A (x) = (x–a_U)/(a_M–a_U) for each x ∈ (a_M, a_U] , and A (x) =0 for each x ∈ (– ∞ , a_L) ∪ (a_U, ∞) , then Ã is called as a triangular fuzzy number and denoted as Ã=(a_L, a_M, a_U) for short [33]. Throughout this research, all fuzzy numbers are considered to be fuzzy triangular numbers.

The decision makers in the firm and the providers can build the triangular possibility distribution of composite requested QoS ${\tilde{q}}_{j} = (q_{jL}, q_{jM}, q_{jU}) \forall j \in J$ and telecom providers’ promised right hand side values ${\tilde{Q}}_{i} = (Q_{iL}, Q_{iM}, Q_{iU}) \forall_{i} \in I$ (see Fig. 2) based on the three noticeable data values as follows [53]:

The most possible value (q_jM and Q_iM) that is definitely a member of the set of available values (normalized possibility degree = 1).

The optimistic value (q_jL and Q_iL) , which has a very low likelihood of being the member of the set of available values (normalized possibility degree = 0).

The pessimistic value (q_jU and Q_iU) , which has a very low likelihood of being the member of the set of available values (normalized possibility degree = 0).

The approach proposed in this paper transforms these imprecise inequalities into a crisp one using the fuzzy ranking concept that is proposed by [28 , 47]. This concept has been applied in many different areas such as supply chain planning, aggregate production planning, and modular software systems selection [19 , 53]. To the best of our knowledge, no study has applied this method to a managerial level telecommunications decision problem. In this method, each imprecise constraint, i.e., QoS constraint, which has imprecise parameters both on the left-hand side and the right-hand side, is replaced with three equivalent auxiliary inequality constraints for a given minimum acceptable possibility value φ (0 ≤ φ ≤ 1) specified by the decision maker. Consequently, the supplementary inequality constraint on Equation (3) can be presented as follows: $y_{ij}, q_{jL}^{φ} \leq Q_{iL}^{φ} \forall i \in I, \forall j \in J$ (18) $y_{ij}, q_{jM}^{φ} \leq Q_{iM}^{φ} \forall i \in I, \forall j \in J$ (19) $y_{ij}, q_{jU}^{φ} \leq Q_{iM}^{φ} \forall i \in I, \forall j \in J$ (20)

It is usually emphasized in the literature that the minimum satisfaction level of the fuzzy constraints or the minimum desired degree of feasibility value φ are elicited by the preference concept. Therefore, it is usually determined by the experience and knowledge of the decision maker [19, 48].

3.4 Design of the heuristic algorithm

The pseudo code and flowchart for the proposed heuristic to solve fuzzy constrained mixed integer non-linear programming model (FC-MINLP) is presented in Algorithm 1 and Fig. 4, respectively. The heuristic starts by calculating the cost of acquiring effective unit bandwidths at each bandwidth level (c_ik) using the UnitCost() function as follows: c_ik ← (c_ik/β_ikL_ika_ik). Next, the set of resources (I) is sorted by effective unit bandwidth cost in ascending order (denoted as A) so that the least expensive resources with the least expensive bandwidth levels are listed at the top of the sorted list $(\bar{I})$ . The set of tasks is sorted by task size in descending order (denoted as D). Then the Find Assignable (j, i) function finds the available resource set $({\bar{I}}_{j})$ for each task that can be allocated by checking the QoS constraints: Equations (18), (19), and (20).

Fig.4

An illustration of the detailed logic-flow for the proposed heuristic algorithm.

The initial selection of resources and allocation of tasks into selected resources occurs between lines 10 and 15 in the algorithm. Acquisition of capacity starts with the cheapest resources per effective bandwidth. Then the tasks that are harder to allocate (tasks with a larger size) are assigned first to selected resources, providing that resource is in ${\bar{I}}_{j}$ and has the available unused capacity (UC_i) that is larger than the size of the task. The allocated task is removed from the sorted task list, $\bar{J}$ , by the Remove() function and resources that have at least one allocation are added to the used (or selected) resource list ( $\bar{I}$ _U) by the Add() function. When a resource is selected to acquire capacity, the bandwidth level of the resource is also selected. Mathematically speaking, selected resource i and the corresponding selected bandwidth level k is denoted as x_ik = 1. Moreover, the Add() function also handles the mechanism for preventing selection of other bandwidth levels from the same selected resources by fixing all x_ik’ to zero where k′ = k so that Equation (9) is guaranteed to hold. For and While loops continue until all tasks are assigned to a resource: $\bar{J}$ =∅, i.e., until Equation (6) is satisfied. Figure 5 depicts a sample initial sourcing and allocation scheme.

Fig.5

A sample initial sourcing and allocation scheme.

In Fig. 5, a firm lease three of a total of four available resources in the market. Resource 1, resource 2, resource 3 and resource 4 have two, three, two, and two bandwidth levels, respectively. The firm leases the highest available bandwidth levels from resources 1 and 2, i.e., bandwidth levels two and three, and the lowest bandwidth level from resource 3. The only size-fixed task in Fig. 5 is the task (a), which is allocated to resource 1. After initial assignment, the total accumulated opportunity cost is zero (second summation term in Equation (1)) because all time-fixed tasks are allocated at their upper transmission rates.

Algorithm 2 documents SubRoutine-A () SubRoutine-A () starts with a very high capacity leasing cost and tries to minimize the sum of the capacity leasing cost and the cost of decreasing transmission rates of time-fixed tasks, task crashing or squeezing. The motivation under SubRoutine-A () .is unselecting (closing) the least used and most expensive resource(s) to decrease capacity acquisition costs by employing following strategies: (i) moving (reallocating) already allocated tasks from the least used, most expensive resource(s) to other resources in ${\bar{I}}_{U}$ that have the available capacity, (ii) unselecting (closing) the least used, most expensive resource(s) and selecting (opening) one upper bandwidth level of already-selected resources if it is feasible and economical, and (iii) unselecting (closing) the least used, most expensive resource(s) and selecting (opening) one lower bandwidth level of a resource that could be closed if it is feasible and economical.

SubRoutine-A() starts with calculating the cost of unused capacity ( ${\bar{c}}_{i}^{u}$ ) using the function UsageCost() that is defined as follows: ${\bar{c}}_{i}^{u} = (c_{ik}) / (β_{ik} L_{ik} a_{ik} - {UC}_{i})$ Then, resources in ${\bar{I}}_{U}$ are sorted by their c_ui values in descending order so that the least used, most expensive resources appear at the top of the newly sorted list. As a next step, the total size of all tasks in the first indexed resource, i $\in {\bar{I}}_{U}$ , is calculated and compared to the unused total capacity in the remaining used resources by setting the transmission rates of time-fixed tasks to the minimum. If the available capacity is larger than the total task size to be reallocated, tasks that will be squeezed (lowering transmission rates) are determined based on the smallest opportunity costs. Then the Swap() procedure is called. The details of the Swap() procedure are given in Algorithm 3. Figure 6 presents a successful Swap() routine and resource closing for strategy (i) that is explained in the previous paragraph.

Fig.6

Swap() and resource closing procedure.

In Fig. 6, resource 1 is the least used, most expensive resource among the three resources already leased. Thus, the Close() function removes tasks (a), (b), and (c) from resource 1. In the final allocation scheme, only task (e) crashes (by lowering the transmission rate) to create slack capacity. If the tasks that are in the least used and most expensive resource cannot be reallocated by just incurring opportunity costs, the algorithm tries reallocating the tasks by first finding the used resources that have at least one more higher-bandwidth level with a cost that is less than the cost of the resource to be closed. This part of the heuristic corresponds to strategy (ii) mentioned earlier and is documented between lines 10 and 21 in Algorithm 2. Successful execution of these lines is depicted in Fig. 7.

Fig.7

Resource closing procedure via opening one higher bandwidth level.

Figure 7 shows that resource 1 is closed by opening a second bandwidth level of resource 3. Task (d) is allocated to the newly leased bandwidth level using the MoveTask() function. Compared to Fig. 6, none of the tasks crash, so there is zero opportunity cost.

The final strategy implemented by the heuristic is to find the least used, most expensive resource that has at least one lower bandwidth level. This is done by calling function FindCandidateResource2(). This function attempts to assign all tasks to the remaining used sources plus the new source which has one lower bandwidth level of closed resource. The idea is to reduce the capacity-acquiring cost by reducing the bandwidth level. However, the opportunity cost will increase when total capacity decreases. Figure 8 shows a simple example of this procedure.

Fig.8

Resource closing procedure via opening one lower bandwidth level.

In Fig. 8, the second bandwidth level of resource 1 is closed, and task (b) is reallocated to resource 2. As can be seen, even though the shape of task (a) has changed, the size (area) of task (a) remains the same.

4 Computational experiments

4.1 Test instances

The data set for the computational study is an extended version of the data set used in [25, 49]. For the computational experiments, 540 unique problems with 54 different settings are generated. Each problem denotes 40 resources from four providers. Problems in which each provider offers enough capacity to fulfill the total demand coming from the customers are created. Hence, using this data, it is possible to analyze under which conditions customers switch providers. As given in Table 4, four parameters (pricing structure, tightness, number of tasks per resource per provider, and the ratio of the number of time-fixed tasks to the number of size-fixed tasks) are identified for the problem sets.

Table 4
Experimental parameters

Parameters Number of levels

Pricing Type 3

Tightness 3

# Tasks Per Resource 3

# Time-Fixed Tasks/ # Size-Fixed Tasks 2

# Problem Types 54

Parameters	Number of levels
Pricing Type	3
Tightness	3
# Tasks Per Resource	3
# Time-Fixed Tasks/ # Size-Fixed Tasks	2
# Problem Types	54

Three pricing scenarios as given in [25, 49] are used. In the first one, a random cost structure is used in which all resource acquisition costs are randomly generated only to assess the performance of the heuristic. In parallel cost curve cases, Provider A is a dominant provider, the least expensive, in all levels of bandwidths (Fig. 9(a)). In intersecting cost curve cases, the price curves of different providers intersect with each other so that at some bandwidths, a provider becomes least expensive (Fig. 9(b), i.e., Provider D with a high bandwidth is the least expensive). Intuitively, the firm will choose the lowest cost. In cases where QoS cannot be fulfilled, the firm will shift to a competitor. The firm will also shift to a competitor when the usage cost of the extra resource with the current provider is no longer cheap.

Fig.9

Provider sensitive pricing cases, (a) parallel, (b) intersecting cost curves.

4.2 Discussion of results

Tables 5 and 6 summarizes the effect of pricing type. When a volume-based discount is applied, the number of resources used and the total cost decreases. When the whole problem set is considered, on average 11.5% reduction in total cost is observed. The average reduction is 8.7% for parallel cost curve cases and 14.6% for intersecting cost curve cases. When the maximum discount offered by providers is considered as 10% (with a 0.90 discount factor), customers in intersecting cost curve cases get more value (less total cost) by acquiring large size resources (having high bandwidths). In this situation, the number of tasks that may incur opportunity costs will increase, and a smaller number of resources helps to lower the total cost.

Table 5
The effects of pricing on various parameters I

Pricing Discount # Resources Acquired Total Cost # Tasks Crashed

Type Leader No Discount Discount No Discount Discount No Discount Discount

Random A 8.91 7.81 1023.31 922.29 113.15 80.03

B 8.81 7.85 1025.21 921.06 113.43 80.01

C 8.73 7.76 1025.07 921.54 112.41 82.37

D 8.70 7.80 1021.70 920.06 112.05 87.46

Parallel A 8.40 7.05 1885.20 1705.86 184.34 160.62

B 8.32 7.08 1878.45 1713.13 182.53 162.53

C 8.26 7.08 1879.92 1723.82 183.69 159.91

D 8.37 6.88 1870.50 1719.29 184.03 163.80

Intersecting A 7.98 7.03 2349.12 2021.48 202.99 179.93

B 8.07 7.04 2371.69 2034.96 202.18 178.22

C 8.19 7.15 2338.15 2020.83 202.85 175.01

D 8.13 7.09 2399.26 2004.55 200.77 175.71

Pricing	Discount	# Resources Acquired	Total Cost	# Tasks Crashed
Random	A	8.91	7.81	1023.31	922.29	113.15	80.03
	B	8.81	7.85	1025.21	921.06	113.43	80.01
	C	8.73	7.76	1025.07	921.54	112.41	82.37
	D	8.70	7.80	1021.70	920.06	112.05	87.46
Parallel	A	8.40	7.05	1885.20	1705.86	184.34	160.62
	B	8.32	7.08	1878.45	1713.13	182.53	162.53
	C	8.26	7.08	1879.92	1723.82	183.69	159.91
	D	8.37	6.88	1870.50	1719.29	184.03	163.80
Intersecting	A	7.98	7.03	2349.12	2021.48	202.99	179.93
	B	8.07	7.04	2371.69	2034.96	202.18	178.22
	C	8.19	7.15	2338.15	2020.83	202.85	175.01
	D	8.13	7.09	2399.26	2004.55	200.77	175.71

Table 6

The effects of pricing on various parameters II

Pricing	Discount	% Resource Utilization		% Difference for Total Task Size		Run Time (CPU sec)
Type	Leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
Random	A	99.60	99.60	13.28	9.81	4034.87	2076.63
	B	99.58	99.62	13.17	9.87	3762.45	2141.80
	C	99.58	99.57	13.12	9.99	3815.11	1960.30
	D	99.59	99.68	13.34	10.49	3267.21	1997.19
Parallel	A	99.55	99.68	19.47	17.37	2025.42	1088.07
	B	99.55	99.67	19.32	17.56	1795.69	1053.17
	C	99.59	99.67	19.36	17.24	1980.95	1152.01
	D	99.57	99.72	19.42	17.62	1807.84	884.79
Intersecting	A	99.38	99.61	20.68	18.82	1124.59	756.87
	B	99.35	99.64	20.67	18.72	900.44	856.04
	C	99.41	99.69	20.78	18.46	953.55	788.73
	D	99.26	99.62	20.49	18.50	1047.14	729.75

As shown in Tables 5 and 6, when a discount is applied, the number of resources used decreases since fewer resources with a high capacity is enough. Even though, the percent difference for the total task size decreases, the percent of bin utilization slightly increases because task allocation to resources with a high capacity becomes easy and the number of tasks crashing decreases. Furthermore, the heuristic finds the solution in a short time because a smaller number of resources is acquired. Heuristic selects a resource and then tries to assign the tasks in that resource to other resources so that the selected resource is emptied. If it can assign all tasks to other resources by incurring opportunity costs, acquisition costs decrease because the empty resource is not acquired. When fewer resources are acquired, the amount of time (iteration number) to assign tasks to other resources decreases and the running time for the heuristic decreases.

The effects of tightness are given in Tables 7 and 8. As tightness increases from 50% to 90%, the total cost increases because the number of resources acquired increases. When a discount applies, one less resource is acquired on average. When tightness increases, the number of tasks that crash and the percent of change in the total task size increases. However, the percent of bin utilization is not greatly affected.

Table 7

The effects of tightness on various parameters I

Tightness	Discount	# Resources Acquired		Total Cost		# Tasks Crashed
	leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
50%	A	6.67	5.50	1215.56	1092.31	127.49	105.32
	B	6.54	5.53	1206.43	1094.90	123.62	102.95
	C	6.62	5.54	1204.68	1089.26	125.75	98.77
	D	6.49	5.46	1212.82	1083.97	127.45	101.64
70%	A	8.36	7.39	1730.96	1535.68	175.67	136.68
	B	8.41	7.36	1753.77	1547.58	178.29	136.54
	C	8.39	7.42	1724.15	1536.91	177.69	141.31
	D	8.42	7.32	1754.66	1545.55	174.83	144.89
90%	A	10.26	9.00	2311.12	2021.63	197.33	178.58
	B	10.25	9.09	2315.15	2026.68	196.23	181.27
	C	10.17	9.03	2314.31	2040.02	195.51	177.20
	D	10.29	9.00	2323.99	2014.38	194.59	180.42

Table 8

The effects of tightness on various parameters II

Tightness	Discount	% Resource Utilization		% Difference for Total Task Size		Run Time (CPU sec)
	Leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
50%	A	99.58	99.60	14.29	12.09	3123.94	1497.94
	B	99.47	99.66	13.92	11.88	3021.34	1723.87
	C	99.50	99.70	14.07	11.49	3080.57	1547.68
	D	99.48	99.67	14.40	11.88	2735.31	1412.11
70 %	A	99.48	99.67	18.65	15.07	2240.89	1409.76
	B	99.49	99.68	18.80	15.10	1834.71	1273.38
	C	99.56	99.61	18.79	15.42	2035.43	1229.56
	D	99.54	99.70	18.55	15.67	1704.38	1081.52
90%	A	99.46	99.62	20.49	18.85	1820.06	1013.87
	B	99.51	99.59	20.44	19.17	1602.52	1053.76
	C	99.52	99.62	20.40	18.79	1633.61	1123.80
	D	99.39	99.65	20.31	19.07	1682.49	1118.10

The running time decreases with increasing tightness (Table 8). In intersecting cost curve case, when resources with high capacities are acquired with 90% tightness, the total cost decreases and the number of scenarios that use alternative resource acquisitions reduces. Hence, the heuristic running time is short. The same results are observed in parallel cost curve cases.

Tables 9 and 10 summarize the effects of the number of tasks per resource. When the number of tasks per resource increases, the total cost increases by 2.1% and the percent of resource utilization increases. The number of tasks incurring opportunity costs increases with increases in the number of tasks per resource. However, the percent difference for the total task size decreases even when the number of crashed tasks increases. When the number of tasks per resource increases, the number of swapping and task allocation operations in the heuristic increases; hence the running time increases.

Table 9

The effects of # of tasks per resource on various parameters I.

# Tasks Per	Discount	# Resources Acquired		Total Cost		# Tasks Crashed
Resource	leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
100	A	8.29	7.21	1732.22	1530.77	112.40	100.74
	B	8.13	7.11	1739.63	1542.94	111.71	101.29
	C	8.24	7.26	1726.64	1535.10	112.21	100.53
	D	8.23	7.12	1749.81	1536.75	112.94	101.90
250	A	8.57	7.38	1772.86	1568.99	221.26	179.65
	B	8.67	7.54	1777.27	1569.83	220.38	179.21
	C	8.55	7.40	1768.79	1575.69	220.43	177.66
	D	8.58	7.40	1777.84	1559.19	218.30	182.74

Table 10

The effects of # of tasks per resource on various parameters II

# Tasks Per	Discount	% Resource Utilization		% Difference for Total Task Size		Run Time (CPU sec)
Resource	Leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
100	A	99.38	99.52	19.97	17.93	246.09	205.39
	B	99.35	99.55	19.86	18.04	262.57	191.05
	C	99.39	99.54	19.92	17.87	266.91	201.69
	D	99.33	99.56	20.05	18.12	250.47	191.63
250	A	99.64	99.74	15.65	12.74	4543.83	2408.99
	B	99.63	99.73	15.59	12.72	4043.15	2509.62
	C	99.66	99.75	15.59	12.59	4232.83	2399.00
	D	99.62	99.79	15.46	12.95	3830.99	2216.19

Tables 11 and 12 summarize the effects of the ratio of the number of time-fixed tasks to the number of size-fixed tasks (TF/SF) ratio on various parameters. As the ratio increases, the number of resources acquired decreases but the total cost increases; also, the number of tasks incurring opportunity cost and the percent difference for the total task size increase. Hence, both opportunity costs and the running time of the heuristic increase.

Table 11

The effects of TF/SF ratio on various parameters I

TF/SF	Discount	# Resources Acquired		Total Cost		# Tasks Crashed
ratio	Leader	No Discount	Discount	No Discount	Discount	No Discount	Discount
0.5	A	8.42	7.28	1662.19	1488.67	112.73	95.58
	B	8.56	7.40	1683.33	1488.09	108.95	94.29
	C	8.47	7.33	1673.68	1487.19	110.07	90.39
	D	8.42	7.27	1676.41	1475.30	109.58	95.12
1	A	8.50	7.28	1753.17	1548.41	164.50	141.74
	B	8.31	7.22	1752.65	1551.22	163.99	141.66
	C	8.44	7.30	1745.87	1559.94	167.79	142.42
	D	8.41	7.26	1748.02	1548.69	167.13	140.42
2	A	8.37	7.33	1842.28	1612.54	223.26	183.27
	B	8.32	7.36	1839.38	1629.85	225.20	184.81
	C	8.27	7.36	1823.59	1619.06	221.09	184.47
	D	8.37	7.25	1867.03	1619.92	220.15	191.42

Table 12

The effects of TF/SF ratio on various parameters II

TF/SF	Discount	% Resource Utilization		% Difference for Total Task Size		Run Time (CPU sec)
ratio	Type	No Discount	Discount	No Discount	Discount	No Discount	Discount
0.5	A	99.51	99.59	12.84	11.02	1369.56	715.31
	B	99.54	99.64	12.47	11.01	1430.36	685.96
	C	99.47	99.57	12.51	10.60	1447.29	709.27
	D	99.45	99.66	12.40	11.04	1282.15	750.38
1	A	99.53	99.63	17.85	15.68	2657.73	1323.68
	B	99.43	99.66	17.76	15.67	2394.08	1346.96
	C	99.55	99.70	18.18	15.76	2532.70	1451.27
	D	99.52	99.68	18.18	15.63	2017.97	1266.92
2	A	99.49	99.68	22.74	19.31	3157.59	1882.57
	B	99.51	99.62	22.94	19.47	2634.14	2018.08
	C	99.56	99.66	22.57	19.34	2769.63	1740.49
	D	99.44	99.69	22.68	19.94	2822.07	1594.42

Table 13 shows the effect of pricing type on the percent of the providers’ resource usage. When the resource usage share of providers is evaluated, similar results are obtained to those in [25], where the reasons for this usage share are explained in detail.

Table 13

% Resource Usage Share of Providers

Pricing	Discount	Provider A		Provider B		Provider C		Provider D
Type	Leader	X	Y	X	Y	X	Y	X	Y
Random	A	25.16	28.88	24.97	24.03	24.32	23.05	25.55	24.05
	B	24.68	23.53	24.72	28.83	24.00	22.93	26.60	24.72
	C	24.87	24.04	22.97	23.57	27.62	29.26	24.54	23.13
	D	23.94	24.09	25.63	24.19	25.67	23.94	24.76	27.78
parallel	A	54.42	65.61	29.45	24.65	12.42	8.30	3.71	1.45
	B	56.56	57.30	28.60	33.12	11.57	8.69	3.27	0.89
	C	55.27	59.28	28.57	26.47	11.80	12.92	4.36	1.34
	D	55.94	60.35	30.66	28.46	10.19	7.92	3.21	3.26
intersecting	A	0.70	0.00	3.80	0.40	20.23	8.44	75.27	91.16
	B	0.30	0.00	5.20	0.23	20.65	10.52	73.84	89.25
	C	1.52	0.00	3.13	0.52	18.87	9.08	76.49	90.40
	D	0.50	0.00	5.27	0.13	22.09	10.49	72.13	89.39

X=no discount offered, Y = discount offered.

In the parallel cost curve cases, Provider A has most of the share since it offers the lowest cost for all bandwidth levels. In intersecting cost curve cases, Provider D has most of the share since it offers the lowest cost with a high bandwidth range. To examine the volume discount effect, the scenarios where each provider is dominant are analyzed. In the parallel cost curve cases, when Provider A is dominant, it increases its resource usage share from 54.42% to 65.61%, which implies a 20% increase with offering a volume discount; other providers’ shares decrease. When Provider B is dominant, it increases its resource usage share from 28.60% to 33.12%, a 16% increase with offering a volume discount. When Provider B is dominant, Providers C and D lose some market share; however, Provider A increases its market share slightly, 1.3%. When Providers C and D are dominant, the increase in their shares is small, but the market share of Provider A increases. Consequently, in the parallel cost curve cases, providers offering low acquisition cost (like Provider A) and a volume discount will increase their market share and profit. Even though other providers are dominant in their discounts, Provider A will benefit indirectly.

In the intersecting cost curve cases, when Provider D is dominant in the volume discount, its resource usage share increases from 72.13% to 89.39%, a 24% increase, and other providers’ shares decrease. When Provider C is dominant in offering more volume discounts, its market share decreases from 18.87% to 9.08%. Consequently, in intersecting cost curve cases, providers offering low acquisition-cost resources with high bandwidths (like Provider D) and dominating the market with high market share, will benefit from offering volume discounts. Further, if other providers offer volume discounts, Provider D will benefit indirectly. For example, when Provider C is dominant (offering a larger volume discount), Provider D increases its market share 18%. Since Provider D offers high bandwidth resources, a volume discount is still attractive by considering acquisition cost, even if it offers a lower volume discount than Provider C. For a more detailed analysis of the market share of providers, the effects of tightness, number of tasks per resources, and TF/SF ratio are summarized in Tables 14 through 16.

Table 14

The effect of tightness on % share of each provider

Pricing	Tightness	Discount	Provider A		Provider B		Provider C		Provider D
type	Leader	X	Y	X	Y	X	Y	X	Y
parallel	50%	A	59.37	72.03	28.67	21.59	10.15	5.65	1.81	0.73
		B	62.10	60.62	26.95	33.19	9.52	5.37	1.43	0.82
		C	61.56	63.46	27.12	25.10	9.57	10.69	1.75	0.75
		D	61.40	69.20	28.65	24.93	9.29	4.81	0.66	1.06
	70%	A	57.47	68.93	27.99	23.73	10.67	6.39	3.87	0.95
		B	56.42	57.35	30.57	32.20	10.17	10.20	2.84	0.25
		C	55.67	59.39	31.07	27.08	10.37	12.45	2.89	1.07
		D	59.21	61.64	29.49	27.36	7.58	7.02	3.72	3.98
	90%	A	46.43	55.88	31.70	28.62	16.42	12.84	5.45	2.66
		B	51.16	53.92	28.28	33.97	15.01	10.51	5.54	1.60
		C	48.58	54.98	27.53	27.22	15.45	15.61	8.44	2.19
		D	47.22	50.22	33.84	33.09	13.69	11.94	5.25	4.74
Intersecting	50%	A	0.00	0.00	0.74	0.00	10.80	2.28	88.46	97.72
		B	0.00	0.00	0.00	0.00	11.33	4.54	88.67	95.46
		C	0.00	0.00	0.00	0.00	9.38	2.37	90.62	97.63
		D	0.00	0.00	0.00	0.00	12.12	2.23	87.88	97.77
	70%	A	0.00	0.00	0.93	0.48	24.22	6.22	74.85	93.29
		B	0.46	0.00	4.41	0.00	22.97	8.09	72.15	91.91
		C	0.57	0.00	0.96	0.00	23.08	6.49	75.38	93.51
		D	0.59	0.00	3.06	0.00	27.88	10.08	68.47	89.92
	90%	A	2.09	0.00	9.74	0.70	25.66	16.82	62.51	82.48
		B	0.45	0.00	11.19	0.68	27.66	18.94	60.69	80.38
		C	3.98	0.00	8.43	1.55	24.13	18.39	63.46	80.07
		D	0.92	0.00	12.76	0.38	26.28	19.14	60.04	80.48

X=no discount offered, Y = discount offered

In the parallel cost curve cases in Table 14, when Provider A is dominant, resource usage share increases with volume discount by 20% with 50% tightness. However, the rate of resource usage decreases with increasing tightness. When Provider A offers a volume discount and tightness increases from 50% to 90%, its resource usage share decreases from 72.03% to 55.88%. When Provider B is dominant, resource usage share is only slightly affected. However, when Providers C and D are dominant, their resource shares increase with increasing tightness. The increment caused by tightness does not affect resource usage share when a volume discount is offered.

In the intersecting cost curve cases in Table 14, when Provider D is dominant, even though the rate of resource usage shares increases with increasing tightness (11.25% with 50% tightness and 34% with 90% tightness), resource usage shares decreases. Provider D has a 97.77% market share when tightness is 50%; however, its share decreases to 80.48% when tightness is 90%. When Provider C is dominant, as tightness increases its resource usage share also increases. The increment caused by tightness does not provide an increase in usage shares when a volume discount is offered. The effects of the number of tasks per resource on providers’ percent of resource usage share are examined in Table 15; however, the effect is somewhat small.

Table 15

The effect of average # of task per resource on % share of each provider

Pricing	# Tasks Per	Discount	Provider A		Provider B		Provider C		Provider D
Type	Resource	Leader	X	Y	X	Y	X	Y	X	Y
	100	A	53.15	65.44	28.13	23.93	14.04	9.10	4.68	1.54
		B	55.67	56.34	30.33	33.47	10.70	9.36	3.29	0.83
		C	54.47	59.32	28.45	25.44	11.52	13.19	5.56	2.05
		D	55.50	60.85	30.85	27.80	10.55	8.34	3.10	3.01
parallel	250	A	55.70	65.78	30.77	25.36	10.79	7.50	2.74	1.36
		B	57.45	58.25	26.87	32.76	12.43	8.02	3.25	0.96
		C	56.08	59.24	28.70	27.49	12.07	12.65	3.16	0.62
		D	56.39	59.86	30.47	29.12	9.83	7.51	3.32	3.51
Intersecting	100	A	0.45	0.00	4.70	0.00	19.61	7.99	75.24	92.01
		B	0.30	0.00	5.66	0.45	21.11	12.18	72.93	87.37
		C	1.10	0.00	4.21	0.36	19.38	8.97	75.31	90.66
		D	0.40	0.00	6.22	0.00	21.51	10.63	71.87	89.37
	250	A	0.94	0.00	2.91	0.79	20.84	8.90	75.30	90.31
		B	0.31	0.00	4.74	0.00	20.20	8.87	74.75	91.13
		C	1.93	0.00	2.05	0.67	18.35	9.19	77.66	90.14
		D	0.61	0.00	4.32	0.25	22.68	10.34	72.39	89.40

X=no discount offered, Y = discount offered

Table 16 summarizes the effects of the TF/SF ratio on the providers’ percent of resource usage share. In the parallel cost curve cases in Table 16, when Provider A is dominant, as the TF/SF ratio increases, the increase in the resource usage rate would decrease share from 23.94% to 18.62%. Hence, the resource usage share also decreases (from 68.59% to 62.7%). In the intersecting cost curve cases in Table 16, when Provider D is dominant, as the TF/SF ratio increases, the increase in the resource usage rate increases share from 19% to 32%. However, when a volume discount is offered, the change in resource usage share is small. When no discount offered and the TF/SF ratio is low, the resource usage share is high.

Table 16

The effect of TF/SF ratio on % share of each provider

Pricing Type	TF/SF ratio	Discount	Provider A		Provider B		Provider C		Provider D
		Leader	X	Y	X	Y	X	Y	X	Y
		A	55.34	68.59	29.01	21.20	12.85	8.93	2.80	1.28
	0.5	B	58.95	56.78	27.92	34.00	10.86	8.96	2.27	0.25
		C	55.25	57.64	28.84	28.71	11.99	12.08	3.92	1.56
		D	54.88	59.16	31.66	28.79	9.80	7.93	3.65	4.12
Parallel	1	A	55.06	65.52	29.45	26.06	11.58	7.03	3.92	1.39
		B	54.60	56.55	29.79	34.85	12.02	7.74	3.59	0.86
		C	55.37	61.73	29.40	24.82	10.44	11.93	4.80	1.52
		D	55.40	59.43	31.97	29.30	9.61	8.05	3.03	3.22
	2	A	52.88	62.73	29.90	26.68	12.82	8.93	4.41	1.67
		B	56.13	58.57	28.10	30.50	11.83	9.37	3.95	1.56
		C	55.20	58.46	27.48	25.88	12.95	14.74	4.36	0.92
		D	57.55	62.48	28.35	27.29	11.16	7.78	2.95	2.45
	0.5	A	0.00	0.00	2.20	0.48	15.93	6.14	81.87	93.37
		B	0.00	0.00	4.20	0.00	19.30	8.47	76.50	91.53
		C	0.96	0.00	2.99	0.86	17.99	7.48	78.06	91.67
		D	0.59	0.00	3.52	0.00	18.95	8.19	76.94	91.81
Intersecting	1	A	0.00	0.00	3.91	0.70	23.87	9.70	72.22	89.59
		B	0.45	0.00	4.55	0.29	20.84	9.17	74.16	90.54
		C	2.45	0.00	2.61	0.43	17.71	11.09	77.23	88.48
		D	0.42	0.00	3.83	0.38	23.74	12.31	72.01	87.31
	2	A	2.09	0.00	5.30	0.00	20.89	9.48	71.73	90.52
		B	0.46	0.00	6.86	0.39	21.82	13.93	70.85	85.67
		C	1.14	0.00	3.79	0.26	20.91	8.68	74.17	91.06
		D	0.50	0.00	8.47	0.00	23.60	10.95	67.43	89.05

X=no discount offered, Y = discount offered

4.3 Performance comparison with the benchmark meta-heuristic

In this section, an extensive comparison of the proposed technique with a GA is carried out to show that our proposed Heuristic outperforms a state-of-the-art general-purpose metaheuristic approach such as GA.

The solution approach offered in Kasap et al. [26] is modified for the comparison. As they mentioned, the tasks are assigned to resources by using GA approach (in other words resource selection) in the modified meta-heuristics. After assignment of tasks to the resources, start time and transmission rate of each task (in other words task allocations) are handled by using the idea of the proposed Heuristic via considering volume discounted pricing. Since one of the meta-heuristics require too much iteration with an excessive amount of running time as given in [26], the other heuristic in [26] has been modified for the comparison purpose.

Tables 17 and 18 summarize the effect of pricing type on the GA approach. Similar to Tables 5 and 6, when a volume-based discount is applied, the number of resources used and the total cost decreases. The proposed Heuristic performs better than GA based approach since the number of resources used and the total cost increases in GA based approach as given in Tables 5 and 17. To compare the proposed Heuristic and GA approach, Table 19 presents % changes for the number of resources used, the total cost and bin utilization in the proposed Heuristic results when GA approach used. In the parallel and intersecting cost curve cases when volume discount applied, the total cost increases around 36% and 40% respectively when in GA approach. Hence, the number of resources used increases 40% in both cases.

As the number of resources used increases in GA approach, bin utilization decreases around 25% as given in Table 19. Although runtimes decrease in GA approach, runtimes for proposed fuzzy heuristic are still acceptable since supplier selection problem is a strategic decision that is not executed on a daily basis

Table 17
The effects of pricing I (with GA approach)

Pricing Discount # Resources Acquired Total Cost # Tasks Crashed

Type Leader No Discount Discount No Discount Discount No Discount Discount

Random A 18.76 9.63 1411.78 1270.87 180.51 183.33

B 18.63 9.69 1433.43 1278.22 180.54 184.35

C 18.71 9.61 1418.96 1265.60 180.44 183.42

D 18.46 9.66 1407.84 1272.56 180.90 183.59

Parallel A 19.31 9.81 2557.14 2324.79 180.12 183.59

B 18.67 9.93 2541.76 2352.31 181.41 183.94

C 18.79 9.78 2532.37 2348.62 179.89 183.04

D 19.05 9.60 2539.79 2326.19 181.31 185.33

Intersecting A 18.88 9.88 3746.92 2827.80 182.24 184.55

B 19.05 9.92 3870.73 2847.70 178.97 182.59

C 18.69 10.12 3693.32 2867.96 183.01 184.65

D 18.78 10.05 3761.84 2815.67 180.19 182.61

Pricing	Discount	# Resources Acquired	Total Cost	# Tasks Crashed
Random	A	18.76	9.63	1411.78	1270.87	180.51	183.33
	B	18.63	9.69	1433.43	1278.22	180.54	184.35
	C	18.71	9.61	1418.96	1265.60	180.44	183.42
	D	18.46	9.66	1407.84	1272.56	180.90	183.59
Parallel	A	19.31	9.81	2557.14	2324.79	180.12	183.59
	B	18.67	9.93	2541.76	2352.31	181.41	183.94
	C	18.79	9.78	2532.37	2348.62	179.89	183.04
	D	19.05	9.60	2539.79	2326.19	181.31	185.33
Intersecting	A	18.88	9.88	3746.92	2827.80	182.24	184.55
	B	19.05	9.92	3870.73	2847.70	178.97	182.59
	C	18.69	10.12	3693.32	2867.96	183.01	184.65
	D	18.78	10.05	3761.84	2815.67	180.19	182.61

Table 18

The effects of pricing II (with GA approach)

		% Resource utilization		% Difference for total task size		Run time (CPU sec)
Pricing type	Discount leader	No discount	Discount	No discount	Discount	No discount	Discount
Random	A	79.45	75.29	15.31	16.43	167.55	143.14
	B	78.51	75.07	15.47	16.63	163.28	144.82
	C	79.49	75.53	15.41	16.52	162.37	145.27
	D	79.60	75.22	15.40	16.60	166.79	140.85
Parallel	A	78.21	73.75	15.43	16.56	136.35	122.90
	B	78.27	73.28	15.64	16.66	137.74	123.75
	C	78.46	73.68	15.35	16.47	137.26	124.85
	D	78.11	73.96	15.62	16.79	136.95	145.40
Intersecting	A	79.05	74.79	15.46	16.60	133.37	122.91
	B	77.87	74.96	15.17	16.51	174.79	123.59
	C	78.58	74.46	15.54	16.67	136.26	124.82
	D	78.70	74.25	15.30	16.39	134.98	156.13

Table 19

% Change in the proposed Heuristic results when GA approach used

		# Resources acquired		Total cost		% Resource utilization
Pricing type	Discount leader	No discount	Discount	No discount	Discount	No discount	Discount
Random	A	110.55	23.30	37.96	37.80	-20.23	-24.41
	B	111.46	23.44	39.82	38.78	-21.16	-24.64
	C	114.32	23.84	38.43	37.34	-20.17	-24.14
	D	112.18	23.85	37.79	38.31	-20.07	-24.54
Parallel	A	129.88	39.15	35.64	36.28	-21.44	-26.01
	B	124.40	40.25	35.31	37.31	-21.38	-26.48
	C	127.48	38.14	34.71	36.25	-21.22	-26.08
	D	127.60	39.53	35.78	35.30	-21.55	-25.83
Intersecting	A	136.59	40.54	59.50	39.89	-20.46	-24.92
	B	136.06	40.91	63.21	39.94	-21.62	-24.77
	C	128.21	41.54	57.96	41.92	-20.95	-25.31
	D	131.00	41.75	56.79	40.46	-20.71	-25.47

Table 20 shows the effect of pricing type on the percent of the providers’ resource usage for the GA approach. When the resource usage share of providers is evaluated, similar results are obtained to those presented in Table 13. In the parallel cost curve cases, Provider A has still most of the share, however, its share decreases 5% and 20% considering with and without volume discounts compared to proposed Heuristic respectively. In intersecting cost curve cases, Provider D has still most of the share, however, its share decreases 7% and 40% considering with and without volume discounts compared to proposed Heuristic respectively.

Table 20

% Resource Usage Share of Providers (with GA approach)

		Provider A		Provider B		Provider C		Provider D
Pricing Type	Discount Leader	X	Y	X	Y	X	Y	X	Y
Random	A	25.86	29.51	22.81	23.87	25.40	23.01	25.93	23.62
	B	24.27	22.79	24.83	28.70	25.29	23.49	25.62	25.03
	C	24.60	24.83	24.51	23.64	26.01	27.87	24.89	23.66
	D	25.31	23.63	24.79	25.42	24.03	24.16	25.87	26.79
Parallel	A	41.87	62.60	31.97	26.47	18.44	8.74	7.73	2.19
	B	43.58	55.25	31.13	33.70	18.00	9.95	7.29	1.10
	C	43.00	56.42	30.59	27.85	18.12	14.14	8.29	1.59
	D	42.55	57.74	32.11	29.71	18.00	8.97	7.34	3.58
intersecting	A	2.05	0.00	11.72	0.50	42.26	15.05	43.98	84.46
	B	1.98	0.18	12.68	0.37	43.13	16.07	42.21	83.38
	C	2.16	0.28	11.01	0.72	42.28	15.88	44.55	83.12
	D	1.55	0.00	12.24	0.25	42.92	17.11	43.28	82.64

X=no discount offered, Y = discount offered.

5 Conclusions and future research

5.1 Summary and contributions

In this article, a cost minimization model for resource acquisition and task allocation problems in the telecommunication networks under uncertainty is studied. The most important contributions of the study are:

Modeling and solving the uncertainty rising due to the network infrastructure and vagueness in users’ QoS requirements by fuzzy optimization techniques. Thus, this is one of the first studies investigating effects of fuzziness in QoS on selection and task allocation policies of firms from a managerial standpoint.

Modeling and analyzing the effect of discount policies from the customer (firm) side rather than from the provider’s standpoint in telecommunications.

Solving a hard optimization problem by the developed effective heuristic by exploring the underlying combinatorial structure of the model developed.

The performed extensive computational analyses on the different input parameters conclude that:

When a volume discount is offered, the number of resources acquired and the total cost decreases. On average, there is an 11.5% reduction in the total cost, 8.7% for parallel cost curve cases and 14.6% for intersecting cost curve cases.

When the maximum discount amount offered by providers to be considered as 10% (with a 0.90 discount factor), customers get more value (less total cost) by acquiring large size resources (having high bandwidths) in intersecting cost curve cases.

In parallel cost curve cases, providers who offer low acquisition costs increase their market shares and their profits by offering volume discounts. For example, when Provider A is dominant (offering a larger volume discount), it increases its resource usage share 20% by offering a volume discount, while other providers’ shares decrease. Even if other providers are dominant in volume discounting, Provider A will benefit indirectly.

In intersecting cost curve cases, providers who offer low acquisition-cost resources with high bandwidths benefit from offering a volume discount and dominate the market with high market shares. For example, when Provider D is dominant, its resource usage shares increase 24% by offering a volume discount, while other providers’ shares decrease. Further, if other providers offer a volume discount, Provider D benefits indirectly. When Provider C is dominant (offering a larger volume discount), Provider D increases its market share 18%. Since it offers high bandwidth resources, a volume discount is attractive, considering unit acquisition costs, even Provider D offers a lower volume discount than Provider C.

5.2 Limitations and future research

Even though the proposed solution heuristic performs quite well and provides essential insights under test scenarios, it may be improved several directions. For example:

Like most of the combinatorial optimization problems, solving FC-MINLP becomes more demanding in terms of running time when the number of tasks, the number of providers, and the number of bandwidth levels increase. Therefore, combining the suggested heuristic with meta-heuristic methods such as simulated annealing and particle swarm algorithms may alleviate this problem. In this case, meta-heuristics might be used for selecting providers and discount levels and the suggested heuristic might be used for solving only task allocation problem.

Although it is usually sufficient to express QoS levels by TFN numbers, improving the proposed heuristic to be capable of handling general fuzzy numbers would make the solution methodology more applicable. In this case, a small change in transformation of fuzzy constraints into equivalent deterministic forms would be sufficient to apply the developed solution heuristic.

It would also be worthwhile to study the effects of different fuzzy ranking concepts that are used for converting imprecise inequalities into crisp inequalities. In this direction, to change the context of FindAssignable(j, i) function would be sufficient to apply the rest of algorithm without any alteration.

Investigating the effects of other discount schemes such as quantity or incremental discounts on provider selection would be an exciting future research direction. However, integrating different discounts schemes would lead to different optimization models and would require changes in both mathematical model and solution heuristic.

Studying a dynamic online problem where tasks are allocated as they arrive would be an interesting contribution. In this direction, using online bin packing algorithms from literature may provide good starting point to find a lower bound for total cost.

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I	The ordered index set of resources
J	The ordered index set of tasks
A_T, A_S	The index set of tasks with fixed transmission time and fixed size respectively where A_T∩ A_S = ∅ and A_T ∪ A_S = J

Volume discount strategies for the provider selection problem in telecommunications under uncertainty

Abstract

Keywords

1 Introduction

1.1 Problem definition and contributions

2.1 QoS and fuzzy QoS

2.2 Discounting Policies

3 Proposed mathematical model with fuzzy constraints and solution algorithm

Table 1 The sets used in problem formulation I The ordered index set of resources J The ordered index set of tasks A T , A S The index set of tasks with fixed transmission time and fixed size respectively where A T ∩ A S = ∅ and A T ∪ A S = J

4.1 Test instances

Table 4 Experimental parameters Parameters Number of levels Pricing Type 3 Tightness 3 # Tasks Per Resource 3 # Time-Fixed Tasks/ # Size-Fixed Tasks 2 # Problem Types 54

5.1 Summary and contributions

5.2 Limitations and future research

References

Table 1
The sets used in problem formulation

I The ordered index set of resources

J The ordered index set of tasks

A_T, A_S The index set of tasks with fixed transmission time and fixed size respectively where A_T∩ A_S = ∅ and A_T ∪ A_S = J

Table 4
Experimental parameters

Parameters Number of levels

Pricing Type 3

Tightness 3

# Tasks Per Resource 3

# Time-Fixed Tasks/ # Size-Fixed Tasks 2

# Problem Types 54