OPTIMAL PRICING FOR COMPETITIVE SERVICE

International Journal of Innovative Computing, Information and Control Volume 7, Number 6, June 2011

c ICIC International ⃝2011 ISSN 1349-4198 pp. 3171–3191

OPTIMAL PRICING FOR COMPETITIVE SERVICE FACILITIES WITH BALKING AND VEERING CUSTOMERS Ali Pahlavani and Mohammad Saidi-Mehrabad Department of Industrial Engineering Iran University of Science and Technology Narmak, Tehran 16844, Iran { pahlavani; mehrabad }@iust.ac.ir

Received August 2009; revised February 2010 Abstract. This paper focuses on determining the optimal pricing strategy for a service providing firm with multiple facilities. The market is involved with some other competing firms providing the same service. The firm and its competitors might devise three different pricing policies as uniform mill pricing, distinct mill pricing and delivered pricing. Customers choose the facilities probabilistically based on their perceived utility to being served. After they received at a congested facility, some of them balk from entering the queue and go directly to another facility rather than coming back to their origins. As a result, the arrival rate for a facility is varying on the level of congestion in all facilities. This situation requires us to find equilibrium point of facilities’ arrival rates. For this purpose, a heuristic procedure is developed. The resulted non-linear and complex model is proposed to solve using a hybrid algorithm of Particle Swarm Intelligence and Differential Evolution, two continuous approximate optimization algorithms. An illustrative example is given to show the model’s applicability and the algorithm’s efficiency. Different scenarios in terms of sensitivity analysis are also mentioned to capture managerial insights. Keywords: Price optimization, Competition, Congestion, Balking with veering, Indirect demand capture

1. Introduction. In a competitive market, players should make a set of critical decisions which help them in capturing further demands or purchasing power of customers. For a service providing firm with multiple facilities, along with an appropriate facility location setting [1, 2] and quality improvement, a decision should be made on its services’ prices. For this purpose, it is required to understand customers’ behavior and their utility factors. It is clear that, congestion in terms of waiting time or occupancy level is an important consideration of quality when a customer decides to buy from a service facility. Each customer might be deterred by the facility’s congestion level and might balk upon arrival. Therefore, the effective demand for a facility’s service is strictly sensitive to its level of congestion. Clearly, congestion becomes a crucial component of revenue management and it must be especially regarded when the firm determines its pricing policies [3]. The role of congestion is usually disregarded in most part of pricing models. They rely on the common economic intuition that lowering the price increases the demand and vice versa. In more realistic situations, however, pricing strictly affects congestion level. This may exert a different direction to the firm’s gains. In particular, once the price is lowered, initially the number of arriving customers will be increased. This will raise the level of congestion such that it becomes unbearable to future customers and probably most of them will be lost. Thus, the really-captured demand would be less than expected. Conversely, once the price is increased, the number of arriving customers will be reduced. 3171

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This makes congestion bearable for the next coming impatient customers and capturing them will compensate partially the lost customers due to price increment. Therefore, the congestion becomes an important aspect of the firm’s pricing strategy. For a firm designed to determine its pricing strategy, initially a distinction should be made between static and dynamic pricing. The dynamic pricing through which charged price varies with the level of congestion are considered by some authors [4-10]. It has been shown that with dynamic pricing, the firm incurs implementation and also image costs which could not be covered by a small extra gain obtained through the implemented scheme and static prices are better for a specific time-period [11]. Besides the general pricing policy, diversification of prices should also be determined. This becomes important when the potential customers differ in their sensitivity to congestion. Normally, firms use either a mill price or a delivered price policy, depending on product or service specification and market conditions. When a firm sets a fixed price at its door and customers pay for transportation, we deal with mill pricing. And when the firms set specific prices at the customers’ door and pay for transportation we deal with spatial price discrimination or delivered price. Although delivered pricing policy has been extensively studied in competitive service pricing models [12], the papers studying mill pricing are rare. In [13], price competition is studied for two types of pricing policies in a market with preestablished facility locations. The authors discussed the existence and determination of equilibrium prices in a general location space. There are also many existing works on pricing in queuing systems. However, a large part of them either considers pricing as a tool for controlling the congestion or studies how queuing costs affect pricing policies and capacity decisions. A work by Naor [14] was the first that studied quantitatively the effect of imposing a toll on arriving customers on congestion level. Since then, several authors studied different approaches to model the effects of toll charges on willingness to wait [15-17]. Customers’ diversification and priority considerations in pricing decisions are also studied [18, 19]. A comprehensive survey on the literature of toll charging models in queuing systems is presented by Printezis [3] who developed a pricing model for a profitmaximizing facility with single queue and multiple servers. He also studied the possibility of price discrimination for two classes of customers. Similarly in a work by Caro and Simchi-Levi [11] the optimality conditions for static discriminatory pricing for diversified customers are studied in a loss network-a special type of queuing system with no buffer. Some authors consider both capacity and price to be optimized simultaneously in order to control congestion levels and consequently improve performance measures [20-23]. Toll setting problem [24] in transportation systems is also a related problem to our considered field. By this problem the operator of a transportation network aims to maximize its profit by imposing prices for its roads using. The users of the network rationally react to the imposed prices. If congestion effects are considered in toll setting problem, it becomes a special case of this work. Along with the models developed for controlling congestion level by imposing tolls, there are other viewpoints on pricing. In [25], a simultaneous optimization model is developed for determining the optimum price and service rate in order to maximize the firm’s profit. It is assumed that demand is elastic to both price and service quality measured by waiting times. In [26] a static pricing problem is developed for a single server facility. The results establish the relation of the optimal price with respect to the system parameters. They assume that customers join the queue if the price charged is less than or equal to a cut-off point and there is room in the waiting area. In a generalized paper [27], the same authors studied optimality conditions for a large class of blocking systems with general arrival and service processes. They investigate how optimal prices change with changes in the size of the waiting room and service rate. In his dissertation, Maoui [28] analyses customers’ balking and reneging behaviors in receiving

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services from a monopolist to determine the optimal static and dynamic pricing policies that maximize the long-run average profit per unit time. A model developed in [29] applies the losses resulted from balking and reneging customers in profit function for adjusting prices in different periods of time. Although the two above studies consider customers’ balking and reneging reactions in queuing systems, they assume that the balked or reneged customers are lost, i.e., they return to their origins. This is not the case for competitive services especially for essential ones. In these situations, the impatient customers who balk from entering go directly to another facility with the aim of receiving the same service in shorter time, i.e., they veer from their initial decisions. To the best of our knowledge, there is not any work studying this behavior in a competitive service facility planning model. In terms of contributions, opposed to most of the literature that focuses on monopolistic pricing of a single facility, this paper provides a price optimization model for a multifacility firm competing for market share with other firms. The model enables us to determine the better pricing policy and also the optimal price vector. It studies an evident reaction of customers encountered by congestion. In this case, we deal with a different concept as indirect demand capture. This needs to find facilities’ arrival rates equilibrium emerged in the optimization model as a complex non-linear system of equations. We present a procedure for solving this system. Furthermore, the model formulates customers’ choice behavior in a different manner compared to the common approach of the literature. In the most part of literature, it is modeled in a deterministic approach which is often called full capture. In this case, the whole demand of a customer is served by a single facility. A more realistic model implying changing mood of customers is the probabilistic approach. In this case, customers distribute their purchasing power among different facilities according to probabilities based on a utility function. This approach is dedicated to Huff [30]. Most competitive location models assume that consumers patronize the closest facility, i.e., the utility function depends only on distance. This is true when differences between facilities are negligible, or in areas where shopping opportunities are few and transportation is difficult [13]. In many cases however, facilities are multiform, i.e., they do differ in other aspects than the mere site where they are located, and customers will take these differences into account in the way they feel attracted to them [31]. Our model formulates the customers’ patronizing behavior in a sequential manner in which customers initially distribute their demand probabilistically based on a utility function depending on distance and offered price and then, decide to patronize a facility based on its congestion level. Regarding the complex structure and non-linearity of the resulted optimization model we cannot solve it using the exact algorithms. Therefore, we develop a hybrid metaheuristic based on Differential Evolution and Particle Swarm Optimization capable of solving small to large-scale models. The rest of the paper is organized as follows. Section 2 describes model development procedure. Section 3 illustrates the model and explains a heuristic procedure for finding the equilibrium point of arrival rates. Section 4 describes the hybrid meta-heuristic algorithm proposed for problem solving. Section 5 gives the experimental results of the model. Section 6 presents some further experimental analyses including sensitivity analysis on the model and finally Section 7 concludes the paper and proposes some issues for future research. 2. Model Development. Suppose that firm A with p fixed facilities in a network is willing to maximize its profit by deciding a pricing strategy given locations and prices of other competitors. The market is formed as a network that includes n nodes (N =

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1, 2, ..., n) as customers origins with demands wi and also as potential facility sites. Let E ⊂ N (|E| = p) be the set of firm A’s facilities and E ′ ⊂ N (|E ′ | = q) be the set of other competitors’ facilities. There is also a set of connecting edges, G each of which indicates the availability of a direct path between two nodes. The network is in a metric space equipped with distance d being the shortest path distance. In the next subsections, we describe main features of the model. 2.1. Customer choice behavior. We try to reflect the reality of customers’ choice behavior in our model by developing a two-step modeling based on three criteria; price, distance and congestion level. At the first step, customers decide probabilistically based on offered price and distance to facility since they don’t know the facilities’ congestion level when they are at their origins. According to [32], the probability that a customer chooses a facility will be as the following: e−υcij xij = ∑ , ∀i ∈ N, j ∈ E ∪ E ′ (1) −υcik e ′ k∈E∪E where xij is the percentage of customers in demand node √ i willing to go to the facility located at j. And υ is a parameter defined as υ = π/σ 6, where σ is the standard deviation in taste of the customers [33]. The dispersion in facility choice increases with decrement of υ. The main indicator of the probability is the cost incurred by customers to being served, cij . It is the sum of offered price and cost of traveling to the facility as follows, cij = pij + t.dij ,

∀i ∈ N,

j ∈ E ∪ E′

(2)

where dij is the shortest path between nodes i and j and t is the traveling cost for a unit of distance and pij is the price offered by facility j to customer i. As the second step, after the customer arrived at a facility, he/she decides to patronize the facility or not based on its congestion level. To reflect congestion effects in the model, we need to define a queuing structure for each facility. 2.2. Congestion modeling. It is assumed that, customers arrive at each facility according to a Poisson process with mean rate λ. Each facility is equipped with m servers all with exponentially distributed service time with mean 1/µ. The system capacity is limited to K customers. Defining occupancy level or the number of customers in the facility as the system’s state, the state probabilities of the resulted queuing system, M /M /m/K, are as the following [34]:  k ρj for k ≤ m  P0j   k!  ρkj P for m ≤ k ≤ K Pkj = (3) m!mk−m 0j     0 for k ≥ K [ P0j = 1 +

K ( ρ )n−m ∑ ρm j j + n! m! n=m+1 m

m ∑ ρnj n=1

ρj =

λj µj

]−1 (4)

(5)

where λj is the arrival rate of facility j. In the case which arriving customers are indifferent to congestion and the system capacity is infinite, the ∑ arrival rate is simply equal to the summation of demand of patronizing customers, i.e., ni=1 γi .xij where γi is the average


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Figure 1. A typical balking function demand generating rate of node i according to Poisson process and xij is defined by Equation (1). The arrival rate of facilities in our formulation is given in the next section. 2.3. Arrival rate. From real situations we know that some of the arriving customers will choose not to wait if they see a long queue, i.e., they balk from entering. We define a parameter β ∈ [0, 1] which accounts for the reduction in initial arrival rates with respect to the queue length faced by the customer. Thus, the arrival rate of node i for facility j will be as the following: λij = β¯ij .γi .xij (6) β¯ij =

K ∑

βk .Pk (λj )

(7)

k=0

where parameter βk is the percentage of the customers willing to enter the queue given that k other customers are attended in the facility. It is defined as { 1 − max(0;k−m) if k 0 and xil ≥ 0 we have yl =

K ∑

i=1

∑p+q j=1

(34)

j=1 j̸=l

rlj ≤ 1, l ∈ E∪E ′ , thus from

∑ k

βk .∂Pk (λl )/∂λl ≤

j̸=l

βk .∂Pk (λl )/∂λl .

k=0

n ∑

γi .xil .(1 − b) − 1 = −a − 1

(35)

i=1

where a is a positive number. Hence yl < 0, ∀l ∈ E ∪ E ′ , and by Definition 3.1, F is Hicksian and by Preposition 3.1, the found arrival rates equilibrium point is unique. 3.1. Equilibrium finding. For finding the equilibrium point in Equation (10), we implement an improved version of the well-known method, fixed point iteration [37]. This procedure that adopts partially the one developed in [38], has the following steps: ′ 1. Based on the current price setting, compute ( ) xij , ∀i ∈ N , ∀j ∈ E ∪ E . (t) 2. Set t = 0 and ⃗λ(t) = 0. Compute Pk λ , ∀k = {0, 1, 2, ..., K}. j

3. Compute (t) φ(λj )

=

n ∑ K ∑

(t) βk .Pk (λj ).γi .xij

i=1 k=0

+

n ∑ K ∑ ∑ l∈E∪E ′ l̸=j

(t)

(1 − βk ).Pk (λl ).γi .xil .rlj

(36)

i=1 k=0

4. Compute a new value

( ) ⃗λ(t+1) = α.⃗λ(t) + (1 − α).Γ ⃗λ(t) , 0 < α < 1

(37)

where Γ is the vector of right hand side functions of Equation (10), φ(.) and α is a user-defined and problem-dependent parameter. 5. Check convergence condition. If it holds, stop with the current solution else set t = t + 1 and go to step 3. Convergence is reached when the value of two successive (t+1) (t) − ⃗λ < ε, where ε is a nonnegative results for ⃗λ become close together, i.e., ⃗λ small real number. Having defined the model and a procedure for equilibrium finding of facilities’ arrival rates, the model can be solved for optimality of charged prices. Since the objective function is strictly non-linear and a non-linear system of equations must be solved for each price setting, the model cannot be solved using standard optimization techniques. Therefore, we implement a hybrid meta-heuristic method for finding the optimum or near optimum solutions.


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4. Hybrid Particle Swarm Optimization and Differential Evolution. The implemented algorithm combines Particle Swarm Intelligence (PSO) [39] and Differential Evolution (DE) [40], two efficient continuous global optimization algorithms. Both PSO and DE work with a population of solutions. PSO is a stochastic search procedure inspired by social behaviors such as bird flocking and fish schooling. The memory structure in PSO retains knowledge of good solutions by all the particles. DE is also a promising new evolutionary algorithm with minimum number of control parameters. It is not only simple, but also performs well on a wide variety of test problems. Regarding their properties, combining the abilities of PSO and DE to get an efficient method seems to be a rational approach. Such combination is developed in [41]. The outline of the implemented hybrid DE-PSO algorithm is as follows: Step 1: Generate randomly POPSIZE random price vectors. They form the initial population g = 0, POP={P1,g , P2,g , . . . , PP OP SIZE,g } Step 2: Update the current personal best (Pibest ) and global best Pbest . FOR each Pi,g , i = 1, 2, . . . , P OP SIZE DO IF f (Pi,g ) > f (Pibest ) then Pibest = Pi,g END IF; IF f (Pibest ) > f (Pbest ) then Pbest = Pibest END IF; END FOR. Step 3: Stopping Criterion IF (g = G), Output the obtained global best solution Pbest and Stop. Step 4: Swarm Evolution FOR each particle Pi,g DO IF (gM ODI! = 0) then Algorithm CUS (Combined updating strategy) Step 4-1: Random Moving Strategy IF (Pbest = Pibest ) then update the particle’s position and speed: xi (g + 1) = Pbest + δ.rand[aj , bj ]nv vi (g + 1) = xi (g + 1) − xi (g) GOTO step 3. END IF; Step 4-2: PSO updating rules vi (g + 1) = w.vi (g) + c1 .r1 .(Pibest − xi (g)) + c2 .r2 .(Pbest − xi (g)) xi (g + 1) = xi (g) + vi (g + 1) ELSE Algorithm DEUS (DE updating strategy) Step 4.1: generate the trial vector yi using the following two DE operations Step 4.1.1: Mutation: generate a vector zi : zi = Pbest + Fi .(Pibest − Pjbest ) if zi ∈ / Ω, select another Fi and GOTO step 4.1.1; Step 4.1.2: Crossover: generate the trial vector yi with zi and xi (g) using the following crossover rule: { j zi if Rj ≤ CR or j = Di uj = xji if Rj > CR and j ̸= Di Step 4.2: Update the particle’s position and speed: xi (g + 1) = yi

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Figure 4. An outline of the market area in the example vi (g + 1) = xi (g + 1) − xi (g) END IF; END FOR. Step 5: g = g + 1; GOTO step 2. The algorithm searches for the optimal solution in G generations. It is generally based on PSO with a randomization mechanism to escape from freezing in population. DE would be implemented once for a cycle of I generations. In the above pseudo-code, Ω denotes the solution space, nv is the number of decision variables and Di is an integer randomly chosen from the set of decision variables index. The superscript j represents the j th element of each solution vector and ai and bj are its lower and upper bounds for Pbest . Rj ∈ (0, 1) is drawn randomly for each j. The parameters r1 and r2 are also chosen randomly from [0, 1]. The algorithm applies a crossover operator which combines the mutated offspring and the current solution to obtain a new solution yi . The set of control parameters includes δ, w, c1 and c2 for CUS routine and Fi and CR for DEUS routine. These parameters could be simply tuned from the literature or implemented experiments. 5. An Illustrative Example. Suppose that a service providing firm (Firm 2) plans to optimize its pricing policy and also price setting for its facilities in a market area formed as a network that includes 20 demand nodes. The firm has just established three facilities. There are also other facilities belonging to another firm competing with us for customers’ purchasing power. The deployment outline of the demand nodes and also facilities is exhibited in Figure 4. Note that all nodes in the network indicate a demand node. A diamond node in the network indicates that a facility belonging to firm 1 has been established at the node. Similarly, the squares indicate the facilities of firm 2 and the circles show the demand nodes with no established facility. The length of available direct paths between the nodes is known and the shortest distance between each pair of nodes is determined. The traveling cost of a distance unit is 0.2 money units. Table 1 gives the demand generating rates of demand nodes while their total demand is assumed to be 1 (wi = 1, ∀i ∈ N ). The column entitled “Included Facilities” indicates that if a facility is established at the node or not. Table 2 gives queuing parameters of the competing firms. The implemented pricing policy by firm 1 is distinct Mill Pricing. Its charged prices are also known.


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Table 1. The specifications of the demand nodes Node Rate Included Facilities Node Rate Included Facilities 1 2.46 Firm 1 11 1.96 — 2 0.41 — 12 2.59 — 3 1.32 — 13 3.02 — 4 3.49 — 14 2.19 Firm 1 5 0.99 — 15 2.14 — 6 3.85 — 16 0.82 — 7 0.98 Firm 2 17 2.82 — 8 1.95 — 18 1.00 — 9 2.60 Firm 2 19 2.53 — 10 3.50 Firm 2 20 0.72 — Table 2. The queuing properties of the firms Number of Facility Nodes Maximum System Service Facilities (No. of Servers) Capacity (K ) (µ) Firm 1 2 1 (3), 14 (3) 18 5 Firm 2 3 7 (2), 9 (3), 10 (3) 18 5 Firm

Minimum chargeable price by the firm is equal to u = 2.5 money units and maximum chargeable price is limited to pmax = 40 units. Different surveys by the social institutes have shown that customers behavior’s uniformity is υ = 0.1. Having defined the problem, we can solve it using the developed algorithm. We coded the algorithm in VB and the computational experiments were carried out on a 1.66 GHz CoreDuo CPU laptop with 1.5 GB RAM. In order to test the algorithm’s efficiency, the results generated by two singular algorithms, PSO and DE are also reported. Table 3 shows the results obtained from applying the algorithms on the problem with different combinations of pricing policies and congestion-modeling approaches. Using the table we can compare our congestionsensitivity approach, the case in which customers return to their origin after balking (full balking) and also the traditional approach without consideration of congestion effects. It also presents a comparison between the three pricing policies. At the first glance on Table 3, we can see that the objective function for the congestion free model is larger than the approaches which consider congestion-sensitivity. This is because that such a traditional modeling that does not account for the congestion effects is strictly optimistic, not a real reflection of service facilities issues. When the congestion is brought to the playing, the impatience of customers becomes important. In this case and with the assumption of full balking, the objective function decreases considerably as the table shows. However, this assumption does not still reflect the reality of congested systems. In our approach, the objective function is smaller than the congestion-free model but larger than the case with full balking. Although customers may balk from a facility of us, they may be also captured by our other facilities. As a result, a smaller percentage of customers will be lost compared to the case of full balking. Figure 5 demonstrates this result through a comparison between the objective functions of three different viewpoints on congestion effects for uniform mill pricing policy. An interesting finding to note from Table 3 is the difference between the outcomes of pricing policies. The better policies are respectively delivered pricing, distinct mill

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Table 3. Computational results of the three proposed algorithms for the three behavior modeling approaches and the three pricing policies

Algorithm

Pricing policy

Uniform Mill PSO Distinct Mill Delivered Uniform Mill DE Distinct Mill Delivered Uniform Mill PSO/DE Distinct Mill Delivered

Without Congestion Obj. CPU Val. Time 234.337 00:00:03 234.337 00:00:16 234.366 00:05:29 234.337 00:00:03 234.337 00:00:16 215.165 00:05:55 234.337 00:00:03 234.337 00:00:16 234.504 00:05:38

Balking Obj. Val. 210.785 210.976 210.875 210.786 210.976 209.22 210.786 210.976 211.104

CPU Time 00:18:13 06:55:57 37:32:33 00:18:46 07:01:18 37:43:33 00:18:26 06:55:57 37:40:42

Balking and veering Obj. CPU Val. Time 229.174 00:48:25 232.073 04:34:34 233.474 46:59:48 229.175 00:45:16 232.146 04:32:12 220.546 101:38:03 229.175 00:49:39 232.147 04:32:10 234.476 91:46:01

Figure 5. Comparison of objective functions pricing and uniform mill pricing with respect to the objective function. The higher is the policy’s flexibility, the larger is the captured customers and consequently the larger is the overall gain of the firm. This outcome however is not a convincing reason for decision makers to implement delivered pricing policy. Some other issues such as the difficulty of customers’ identification, the problem of further amount of needed human resources and the computational issues should be considered to come to a better decision for pricing policy. The final analysis on Table 3 studies the three implemented algorithms. Comparing the results of the algorithms indicates that the hybrid algorithm provides better result than the two singular algorithms with respect to both quality and computational time measures. However, the problem solving for delivered pricing policy is computationally expensive than that for distinct mill pricing and uniform mill pricing. This could be simply reasoned


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Figure 6. Sensitivity of (a) optimal price and (b) profit to the change of firm 1’ charged price regarding the number of decision variables for different policies and the fact that we deal with a continuous solution space. It may be interesting to study the market share of the firm under different viewpoints its entrepreneurs may have on the market. They may decide to increase its coverage over the market or they may aim to gain further profit. We study this situation by assuming different objective functions for the problem. The results of the experiment on the uniform mill pricing policy are depicted in Table 4. Table 4. Comparison of results considering different objective functions Objective function Best Price Captured Demand Revenue Profit

2.495 17.472 20.311

Captured Market Share Revenue Profit Demand 33.885 82.0% 84.556 –79.164 20.461 49.5% 357.508 227.284 17.307 41.9% 351.527 229.175

From Table 4 we can see that the maximum market share is captured in the case of considering captured demand as the objective function. Note that the optimal price in this case is equal to the minimum chargeable price. This is an expectable outcome since further customers are attracted to the facilities. However, the availability of fixed costs leads to a negative profit. The minimum market share is obtained by solving the model using the profit objective function. In this case, the firm offers better services for high potential customers and gratifies them to maximize its gains. As a result the firm cannot support all areas of the market. 6. Further Analysis. In this section, the developed model is further investigated by studying its outcomes in different situations and for various assumptions. To facilitate the perception of analyses, the example of previous section is considered as a test bed. It is assumed that the pricing policy is uniform mill pricing. However, all of the analyses could be simply generalized to the two other policies. 6.1. Competitors’ pricing. As the first experiment we are interested to test the effect of the competitor’s pricing on our gains. For this purpose we assume that some changes are applied to the price vector charged by firm 1, our competitor. The experiment is shown in Figure 6. As it can be seen from Figure 6, both optimal price and profit are varying in the same direction with the change of the competitor’s charged prices. This is an expectable

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Figure 7. Sensitivity of (a) optimal price and (b) profit to system capacity (K ) outcome because we must decrease our prices to maintain overall gains as the competitor lowers its charged price. In contrast, when the competitor charges a higher price, as a good opportunity, both firms can gain further. This will be held while the demand is inelastic to the price. The increasing slope of both curves results from the fact that, when our competitor charges a higher price, we can also raise our price such that not only we wouldn’t lose any part of the before-attracted customers, but also capture those customers disregarded by our competitor. This outcome indicates that, any change in the strategies of our competitors will affect our achievements. Therefore, it is required to revise our strategies. This fact points out an important issue, price equilibrium or the state in which none of the firms is interested to change its prices for further gains. 6.2. Queuing parameters. As discussed before and also illustrated by experiments, congestion plays an important role in spatial planning decisions for service facilities. Therefore, it is rational to expect that the outcome of the model is strictly sensitive to the queuing parameters. Some experiments on queuing parameters can verify this proposition. We will see how the optimal prices and objective function change, as the queuing parameters of the system change. This is similar to the sensitivity analysis performed in [10,28]. At the first, we test the system capacity parameter, K. This experiment is shown in Figure 7. As expected, the smaller the capacity is, the smaller the profit will be. According to Figure 7, the optimal price curve is decreasing because with a smaller price further customers could be captured for a larger capacity. Although the profit is increasing with K, its slope is decreasing. This is because that for some facilities, the assigned capacity is larger than the capacity needed for capturing the whole attracted customers. The next experiment studies the effect of mean service rate, µ on the firm’s achievements. Mean service rate could be accounted for as the productivity of a server. This experiment is shown in Figure 8. From Figure 8, we can see that the optimal price is decreasing because with price decrement, further customers could be attracted to the new capacity created through increasing the mean service rate. As expected, a larger mean service rate leads to better gains. But similar to the system capacity, the slope of profit curve is decreasing. This is because that for some facilities, the assigned mean service rate is larger than the rate needed for serving whole attracted customers.


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Figure 8. Sensitivity of (a) optimal price and (b) profit to mean service rate (µ)

Figure 9. Sensitivity of (a) optimal price and (b) profit to the server numbers at facility 10 An interesting result in the two above experiments is that the optimal price is decreasing with increment of both system capacity and also mean service rate. The only difference between them is on the slope of the price decrement that is larger for the mean service rate than the system capacity. This means that the mean service rate highly affects the system’s performance compared to the system capacity. Another queuing parameter on which the firm’s planners could impose its decisions is the number of servers in the facilities. To analyze this parameter, we assume that the number of servers at the facility located at node 10 is varying between 1 and 10. Figure 9 shows this experiment. According to Figure 9, the best alternative for the number of servers assigned for facility 10 with other parameters remained unchanged is to establish two servers. Assigning more servers to this facility worsens our gains due to incurring irrecoverable servers’ fixed operating costs. 6.3. Customers’ behavior. In the third set of experiments we study the effect of some customer related parameters on the optimal price and the firm’s profit. The first item coming to mind is the vector of demand rates. In the experiment depicted by Figure 10, we change the initial vector of demand rates to follow its effect. As expected, both the optimal price and profit are increasing with the demand rates. Note that this may also happen for our competitors. Figure 11 shows an experiment which examines the model with respect to the customers’ travelling cost.

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Figure 10. Sensitivity of (a) optimal price and (b) profit to customers’ demand rate

Figure 11. Sensitivity of (a) optimal price and (b) profit to customers’ traveling cost

Figure 12. Sensitivity of (a) optimal price and (b) profit to customers’ choice uniformity Both the optimal price and profit increases with increment in the traveling cost. However, this is not a general conclusion for all firms. The influence of this parameter would be smaller for the firms with further presence in the market because their facilities are easily accessible to larger part of customers, i.e., they are lowly affected by the distance parameter. The final experiment examines how the model is sensitive to the customers’ choice uniformity (υ) in Equation (1). This parameter strictly affects customers’ patronizing behavior.


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From Figure 12, we can see that although the optimal price decreases with increment in the customers’ choice uniformity, but the profit decreases first and then increases. Similar to the travelling cost, the degree of influence of customers’ choice uniformity depends on the facilities’ locations. 7. Conclusions and Future Researches. In this paper, we have considered the optimal pricing problem for a service-providing firm with multiple established facilities to maximize its long-run average profit per unit time. The presented model could be applied in a multi competitor market with consideration of different pricing policies. It supports managers in deciding optimal pricing including better policy and optimal price(s). The paper mainly contributes to the customers’ sequential behavior with emphasis on the congestion effects. Their initial decision for patronizing different facilities is based on offered price and travelling cost. In an initially selected facility, customers react to the congestion level by presenting impatience through balking from entering. Some of them will go directly to another facility rather than coming back to their origins. We have called this as veering behavior. Consideration of this behavior has resulted in a complex non-linear system of equations for computing the facilities’ arrival rates. The arrival rate for each facility is constructed from two main parts, direct captured demand from demand nodes and indirect captured demand of customers balked from the congested facilities. Since any fluctuation in the charged price affects congestion levels, an equilibrium point of arrival rates should be found for each pricing setting. A routine has been devised to solve the equilibrium system of equations. A hybrid efficient meta-heuristic algorithm has also been proposed to solve the developed optimal pricing model for different pricing policies. It is based on hybridization of Particle Swarm Intelligence and Differential Evolution. An illustrative example has been presented to show the applicability of the model. Various experiments have been performed to analyze different aspects of the model. The experimental analyses present its better coincidence with rationality and intuition in comparison to the other models developed on similar cases. There are some issues to be considered in future researches. Among them, the most important one is to study the interactions between competitors. Obviously, once a firm devises a pricing policy and sets some prices for its services, its competitors react and try to amend their policies for maintaining competitive advantages (as perceived by Figure 6). So, taking competitors’ reactions into account and finding equilibrium prices should be a serious issue to be considered as a potential research. Due to interactive effect of both distance and price on customers’ patronizing behavior, it will be interesting to consider simultaneously both location and price decisions to be optimized. However this leads to a hard and complex problem. Acknowledgment. The authors would like to thank the reviewers for their helpful comments and suggestions which have improved the presentation. REFERENCES [1] T. Uno, H. Katagiri and K. Kato, An evolutionary multi-agent based search method for Stackelberg solutions of bilevel facility location problems, International Journal of Innovative Computing, Information and Control, vol.4, no.5, pp.1033-1042, 2008. [2] J. G. Cabrera, N. R. Smith, V. Kalashinkov and E. Cobas-Flores, Mathematical model for retail stores location in competitive environments, International Journal of Innovative Computing, Information and Control, vol.5, no.9, pp.2511-2521, 2009.

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