The optimal pricing model in an uncertain and competitive environment : using possibilitic geometric programming approach

Providing internet to the customers is a huge and highly competitive market. Offering internet with a competitive price and high level of quality to the customers is a challenging issue to all server-owners at large. Internet providers usually have a limited bandwidth which should try to allocate it to different services as such to maximize profitability. As a result, they should care about several factors such as the amount of bandwidth allocated to each service, amount of marketing expenditure, selling price, level of system's reliability, and so forth. On the other hand, presence of uncertainty makes it almost impossible to achieve a good estimation on the model's key parameters such as price elasticity. Therefore, in this study, we present the outcome of our investigations as a mathematical model to maximize the profit of a server-owner in an uncertain and competitive environment. The uncertain parameters are taken as a triangular fuzzy numbers. The resulted problem is modeled as a general form of Possibilistic Geometric Programming (PGP). It is evident that the developed model is a highly nonlinear one. As a result, we use CVX to solve it optimally. Finally, the numerical analysis is applied for validation of the model.


INTRODUCTION
In a typical today's competitive market, businesses are faced with a few important and challenging decisions of determining what to sell, how much to sell, and more importantly on which price to sell.
Purposing a reasonable answer for these set of questions is not an easy task since traditional models make various simplifying assumptions in order to arrive at a near optimal solution.As an example, it is assumed that customers' demands are exactly known or systems operate perfectly.However, growing competiveness in today's market has inspired many researchers to study and propose models under more realistic assumptions.
The key point to respond appropriately to this *Corresponding author.E-mail: ragheb.rahmaniani@gmail.comenvironment is defining a strategy which closely follows market conditions and meets the demand.Accordingly, Revenue Management (RM) techniques are also effectively implemented (Bitran and Caldentey, 2003).In general, revenue management problems can be categorized into forecasting, overbooking, pricing, and capacity control (Belobaba, 1987;McGill and Ryzin, 1999).However, pricing is one of the most important applications of RM which has been investigated by many authors during the last few decades (Bitran and Caldentey, 2003;Caoa et al., 2012); Kim and Lee, 1998;McGill and Ryzin, 1999).In line with this subject, Lee and Kim (1993) are regarded as the pioneer in integrating marketing and production models.
In their model, demand is considered as an exponential function of price and marketing expenditures and the production cost is also a function of demand.They also formulated their problem as a Geometric Programming (GP) model.Later on, they incorporated the capacity of a market in their model either fixed or variable (Kim and Lee, 1998).Sadjadi et al. (2005) proposed a GP model to optimize production and marketing planning where demand depends on price and marketing expenditures, and production cost is inversely related to the lot size.Furthermore, they presented a closed form solution for the problem.Fathian et al. (2009) presented a pricing model for electronic products and used the GP dual method to find the global solution for the problem.Considering reliability in optimal pricing models have been another issue dealt with by some researchers.Leung (2007) has investigated an EPQ model with a flexible and imperfect production process in which interest and depreciation costs are depend on the process flexibility and its reliability.He used a generalized geometric-programming solution to solve the model.
One of the most important and practical applications of RM is to determine an optimal strategy for Internet Service Providers (ISPs).Generally speaking, a serverowner pays money for a limited internet bandwidth to a vendor and apportions it to different services in such a way that the summation of all allocated bandwidths to each service does not exceed the purchased bandwidth.The primary concern is to maximize the selling profitability and to increase the market share in a competitive environment.Therefore, mangers require to invest a great deal of money annually to keep service reliability and raise their market share through advertisements subject to some strict constraints, like available budget for marketing.Moreover, they should offer their service with a competitive price compared with the rival's selling price to increase their competitiveness in market and subsequently their market share as well.Meanwhile, internet's technology is under significant and continues developments.Nair and Bapna (2001) considered the structure of ISPs market and stated that demand and supply in this sector is different from other industries, since demand and service happen simultaneously and system must be ready at every moment to serve customers and lacks of overbooking.Novak (2008) discussed a relative cost-based solution for optimizing bandwidth allocations on a campus gateway Internet link.At another work, Novak (2010) considered blocking thresholds for dialup ISPs.Bouhtou and Erbs (2009) investigated the problem of maximizing the revenue of allocating bandwidth in a competitive environment.Youa et al. (2009) developed a constrained nonlinear integer model for bandwidth allocation, in order to maximize the total expected revenues.They also proposed a particle swarm optimization algorithm to solve the problem.
Unfortunately, traditional pricing models do not take the effects of uncertainty into account.For example, it is assumed that model's key parameters such as elasticities are certainly known in advance, whereas, in reality, this assumption is rarely satisfied and in most cases due to changes in the parameters, the decision made may become ineffective (Shah et al. 2012).Therefore, in order to be more realistic, and to respond more appropriately to the growing competitiveness in the global business environment, recently some authors have studied the optimal pricing models with more realistic assumptions.Sadjadi et al. (2010) presented a new marketing and pricing model in the fully uncertain environment where decision variables were fuzzy numbers.They have developed a decision support system to inference vector of decision variables without solving geometric programming problem directly.Bag et al. (2009) considered demand as a fuzzy random variable and the set-up cost, reliability of the production process, and production period are considered as decision variables.The unconstrained signomial GP method is used to determine the optimal decision.
In the present paper, we study a pricing model for ISPs with the consideration of reliability of services given to the customers.We develop the model formulation with the same argument that, demand depends inversely on selling price and directly on marketing expenditure (Lee and Kim, 1993).In addition, the maintenance cost is inversely related to the reliability.As a result, we present a nonlinear mathematical model to find the optimal pricing strategy in an incomplete, competitive, and uncertain environment for a company which provides internet service to the customers.On the other hand, the absence of historical data for services makes it difficult to provide a fair estimation on the necessary parameters such as price elasticity.The resulted problem is formulated as a general form of possibilistic geometric programming (PGP).
The paper is organized as follows.In the next Section, the problem statements and assumptions for the problem in hand are discussed.Crisp and fuzzy mathematical formulations of the problem are presented in Section 3. Solution approach is the discussed in Section 4. Numerical example and sensitivity analysis of the optimal solution are subjects of Section 5. Finally, in Section 6 we draw conclusions and specify future research directions.

PROBLEM STATEMENT AND ASSUMPTIONS
Consider an internet server with limited bandwidth (C ).This bandwidth should be allocated to different services (that is, ).The main concerns are to determine (1) amount of allocated bandwidth to each service and (2) the optimal pricing strategy for each service.The server cannot supply demands at every moment and it may shutdown because of power failure or upgrading software, which causes in losing the demand.To be more precise, if r indicates reliability of system, we loses (1-r) % of demand (that is, (1

totally
(1 ) j j j r D P − ∑ of profit.However, we can reduce the lost sale by achieving to a higher level of reliability.In other words, through substantial investments in several factors such as buying a backup power system or improving server's technology or depending on more sophisticated software, the system's reliability can be considerably increased.In addition, another advantage of higher reliability is to reduce maintenance costs.Therefore, the internet provider is also faced with the decision regarding optimal levels of system's reliability.On the other hand, as a result of intensifying the competition in the market, the server-owner knows that promotional activities and level of reliability of the service nowadays play a more significant role in increasing his profitability and market share.So, he/she needs to make decisions about the amount of investments in advertising programs.On the other hand, amount of available budget for marketing is naturally restricted and the selling price should not be kept greater than ( ) selling price.These perspectives can be translated into a mathematical formulation.In short, we make following decisions: The optimal selling price for each service Level of reliability Amount of investment in advertising programs for each service In such a way that: Maximizes total profit Summation of total allocated bandwidth to different services should not exceed the purchased bandwidth The marketing expenditure should not exceed available budget The selling price be lower than ( ) different service, the proposed crisp model from the server-owner perspective would be as follows: Subject to: Rahmaniani et al. 11567 ( ) Equation ( 1) indicates the objective function, which maximizes total revenue.First term (that is,

∑
) represents total selling profit.Since the server does not perform perfectly, it cannot supply all demands.Therefore, we just earn money from selling % r of total demand.Second term indicates the total marketing expenditure.Finally, the term written out of parentheses is maintenance cost of the server.Equation (2) guarantees that total allocated bandwidth to services do not exceed the purchased bandwidth.Equation (3) shows the constraint on the total budget available for marketing.Constrain (4) ensures that our selling price for each service should not exceed ( ) price.Equation ( 5) restricts the reliability to be at most V. Finally, constraints (6) represent decision variables.The following power function relations are defined for our model.
In Equation ( 7), demand is defined as a decreasing power function of selling price and increasing power function of marketing cost.Similar structures for the demand function are proposed by other researchers (Glickman and Berger 1976;Chen, 2000;Jung and Klein, 2006;Esmaeili and Zeephongsekul, 2010).Generally, by increasing reliability failures greatly reduce.Which, in turn, leads to the maintenance costs drop.As a result in Equation ( 8), maintenance cost is defined as a decreasing power function of process reliability.

Fuzzy pricing model
Due to changes in the business environment, it is not reasonable to make decisions based on deterministic values and we should take the effects of uncertainty into account, while developing a mathematical model for a problem.Therefore, in order to face with the fierce business environment better, we assume that elasticity parameters are not clearly known and, , , ( ) ) Although there are many other approaches to overcome the uncertainty in the parameters, in this article, we use fuzzy approach because, as we were asking the server-owner regarding parameters, he/she would just describe the model's parameters by linguistic variables.In addition, there is no historical information available to estimate the probability distribution of parameters.As a result, using fuzzy approach to overcome the uncertainty seems to be highly appropriate and rational.Accordingly, the proposed model formulation for an ISPs in a fuzzy environment would be as follows.
Subject to: (4), ( 5), and (6) This model is a possibilitic geometric programming and we use fuzzy goal programming method to solve it.Accordingly, an aspiration level like ( , , ) is defined and the objective function is added to constraint via Z % as follows: max Z % (15)

SOLUTION APPROACH
The proposed model can be solved by using the method described by Sadjadi et al. (2005).However for the sake of simplicity, we assume that the internet provider just offers one service (that is 1 J = ).Consequently, with this simplification and also with one additional variable (that is.z) and a constraint, the model can be converted into the following conventional posynomial geometric programming form.
Subject to: The proposed model ( 21)-( 30) is a very difficult GP problem containing 7 degrees of difficulties.Therefore, we use the CVX Modeling System (Grant and Boyd, 2009) which can be implemented on MATLAB software in order to find the optimal solution of the problem.

Numerical experiment
The required parameters to build the model are gathered by interviewing experts.In general, the experts could not indicate exact values for parameters and they just described parameters by linguistic variables.On the other hand, there was no reliable historical information to estimate parameters from.These restrictions motivated us to use fuzzy programming approach to solve the problem.As a result, we implemented the proposed model to find the optimal pricing strategy for particular ISPs with limited bandwidth of 51200 kb/sec which offers just one type of service (that is.

/ sec kb ϕ =
).The server-owner wants to determine optimal selling price, amount of marketing expenditure, and optimal level of server's reliability.Furthermore, total available budget for marketing expenditure is limited to 15000$.Furthermore, the rival's selling price is 160$ and he should not sell our service more than 161.6$ (that is 0.01

ρ =
).For this case, other parameters are

Sensitivity analysis
In the following subsections, we apply sensitivity analysis to demonstrate the effects of change in the input parameters on the optimal values of decision variables in the proposed GP model.Nevertheless, we should not always expect a certain behavior for interdependencies of some decision variables because of high degrees of nonlinearities.

The effect of changes of α% on the optimal decision variables
In order to carry out this analysis, we examined both increasing and decreasing patterns in values of α% .As a result, we increase/decrease values of α% in a range of 0% to 50%.It is observed that the reliability is not much sensitive to these changes.In Figure 1 we have plotted changes in decision variables against increasing the value of α% .We can observe from Figure 1 that as α% increases, the optimal value of all decision variables decreases.Selling price (P) decreases since the price elasticity to demand increases, the server-owner has to decrease his selling price in order to keep the desired market share.
Likewise, by increasing α% , total costs increase, and since the objective function is to minimize total costs, marketing expenditures require to be decreased.On the contrary, as α% decreases, the optimal value of all decision variables would increase.However, decreasing selling price elasticity to demand more than 21% makes the proposed GP model infeasible.Figure 2, illustrates the plot of decreasing effect in α% on decision variables (Figure 5).

The effect of changes of γ on the optimal decision variables
In this part, the effects of changes in γ% on optimal values of decision variables are investigated.We manipulated the marketing elasticity of demand in a range of 0% to 100%.As our computational results indicated, decision variables except of marketing expenditure (M) are not much sensitive to this analysis.It can be noted from Figure 3, as we increase the value of γ% , marketing expenditures decreases (Figure 3), because with the rise in power of M total costs increase and, in order to maximize revenue, marketing expenditures (M) should decrease.On the other hand, as  the power of M decreases its weight in objective function decrease and we can still increase our investment in marketing, while it does not considerably affects total profit.
Increasing pattern Decreasing pattern  Moreover, increasing investment in advertisement directly increases the market share which, in turn increases total profit.It worth noting that, reliability did not show a regular behavior and this is due to the high degree of problem's difficulty (that is 7).

The effect of changes of K on the optimal decision variables
We considered different values for analyzing the effect of changes in K on the optimal solution according to { } 6 1 10 ; 0,1, 2,...,300 K t t = + ∀ ∈ . We observed that an increase in K, causes the optimal product price and, also, marketing expenditures to increase (Figure 4).All in all, K directly impacts all decision variables.Worth noting that, in this analysis the reliability is not also much sensitive.

The effect of changes of V on the optimal decision variables
To carrying out this analysis we increased the value of V in each step by 0.05 with beginning from 0.2, because for values less than V = 0.2 the problem is infeasible, that is; the server should work at least with reliability of 20%.Generally speaking, increasing V causes increasing in decision variables, because with increasing r, earned money from selling service (that is r D P

↑
) increase and maintenance costs decrease.However, selling price is not much sensitive to changes in V.

The effect of changes of B on the optimal decision variables
Finally changing the maximum available budget just affects the marketing expenditure.As we gradually increase the value of maximum available budget for marketing (B) from 1 to 10^5, the marketing expenditure increases, because we are loosing budget constraint.Subsequently, for greater values than 78*10^3$, increasing B does not causes any change in marketing expenditure, since the budget constraint is relaxed.This phenomenon can be observed in Figure 6 CONCLUSION AND FUTURE RESEARCH DIRECTION In this paper, a mathematical model in an uncertain and competitive environment to find the optimal pricing strategy for ISPs problems was investigated.Uncertain parameters in the proposed model were fuzzy numbers with triangular possibility distribution.As a result, model's stability against different real world situations has been increased by far.Moreover, since the server-owner just can describe the model's parameters by linguistic variables and typically there is no historical information to estimate the probability distribution of parameters, using fuzzy approach to overcome the uncertainty was highly appropriate.In addition, we assumed that there are rivals for the service which is offered to customers in market.Consequently, in order to increase the market share, the server-owner should invest a considerable amount of money in advertisement, increasing system's reliability, and more importantly he should not sell his service more than ( ) 1 % ρ + of rival's selling price.By translating these perspectives into mathematical formulation, we presented a possibilitic geometric programming model.Finally, we presented numerical experiments and sensitivity analysis.
In this paper, we assumed that inflation does not affect our pricing strategy, while in reality inflation may increase prices by 30% or more.Therefore, one logical extension to the current research is considering the impact of inflation in the model.Moreover, in order to increase market share, it is logical to gives discount to the customer based on number of hours which they are connected to internet.So, giving discounts to customers can be another appropriate extension to the proposed model in this paper.
customers' demand is optimal.Total demand achieved and total satisfied demand are respectively, [ ] 1601.07,525.30, 98.74 and [ ] 1585.06,520.05,97.75 .And finally, total expected profit and maintenance costs are respectively

Figure 1 .
Figure 1.Effects of increasing in α% on decision variables.

Figure 2 .
Figure 2. Effects of decreasing in α% on decision variables.

Figure 3 .
Figure 3. Effects of changing in γ% on decision variables.

Figure 4 .
Figure 4. Effects of increasing in K on decision variables

Figure 5 .
Figure 5. Effects of increasing in V on decision variables.

Figure 6 .
Figure 6.Effects of increasing in B on M.