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Capacity-constrained multiple-market price discrimination
Computers & Operations Research 39 (2012) 105–111
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Capacity-constrained multiple-market price discrimination Bin Zhang Lingnan College, Sun Yat-Sen University, Guangzhou 510275, PR China
a r t i c l e i n f o
abstract
Available online 5 February 2011
This paper studies a multiple-market price discrimination problem with different markets’ demand elasticity and supply constraints, whereas the markets share a common capacity. We model the problem as a continuous nonlinear knapsack problem, and propose an efficient algorithm for solving the optimal solution after exploring the structural properties of the studied problem. Sensitivity analysis is done for better understanding the behavior of the problem, and numerical results are provided to show the performance of the proposed method. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Price discrimination Nonlinear programming Knapsack problem
1. Introduction Price discrimination has been intensively practiced in sales management when a firm charges different prices for same/ similar products in different markets [1,2]. In sales practice, price discrimination is popular because it often generates more profits over single-price strategy in all markets when the multiple markets have different demand elasticity. Firms in various industries such as airlines, hotels and theaters often face fixed capacity constraints when using price discrimination [3]. Under these circumstances, the shared capacity should be well allocated to different customer groups, and it has important influence on the optimal pricing decisions. The capacity-constrained price discrimination problem has many applications in practice, which has been studied in the last years. Wilson [4] studied deterministic multi-product models by designing an optimal menu of prices and products. Reece and Sobel [3] proposed a diagrammatic approach for solving twomarket case with linear demand model by comparing the marginal costs and marginal revenues. They plotted marginal revenues from two markets and capacity constraint in one figure to find the optimal prices for two markets. This diagrammatic approach cannot be extended for solving multiple-market case, and it cannot deal with market-specific capacity constraints. Baylis and Perloff [5] investigated two-market case through Karush– Kuhn–Tucker (KKT) approach. Bitran and Caldentey [6] modeled the problem as a multiple-product deterministic demand pricing model, and characterized the optimal properties by applying KKT conditions. Shy [7] investigated the problem with linear demand model and without market-specific capacity constraints,
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and solved the problem by solving a system of equations. Malighetti et al. [8] considered multiple-market pricing problem for low-cost airlines in which the demand for air tickets depends on price levels and on the advanced purchase time, and solved the model through KKT approach. Dana and Yahalom [9] studied price discrimination with a resource constraint by explicitly modeling consumers’ types. In this paper, we extend the research on capacity-constrained price discrimination problem through taking into account the following aspects: (1) we consider the different supply costs of products in different markets, as [10] mentioned, standard theory neglected this costs. (2) We model demand in general forms including linear and nonlinear demand model. (3) We investigate the case of multiple markets instead of two-market case. (4) We explicitly model the shared common capacity as a knapsack constraint. (5) We introduce multiple capacity constraints (one for each market) to reflect other production or market restrictions on selling quantities. Although some of these aspects have been studied in literature, up to now, we have not found any publication that has considered all of them. All these generalizations make the optimal pricing decisions more challenging. For example, Mujumdar and Pal [11] showed that the restriction on decision for one market had important influences on the optimal decision for another market by analyzing a two-market pricing problem. We model the capacity-constrained multiple-market price discrimination as a continuous nonlinear knapsack problem. Continuous nonlinear knapsack problems have been intensively studied in last decades, because of their immediate applications in industry and financial management, but more especially for theoretical reasons. Nonlinear knapsack problems have many applications, such as multi-product newsvendor, production and inventory, Euclidean projection, portfolio selection, video-ondemand batching, and optimum allocation in stratified sampling
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B. Zhang / Computers & Operations Research 39 (2012) 105–111
problems [12,13]. Beside their direct applications, nonlinear knapsack problems have been widely studied because of their frequent appearance as ‘core’ subproblems in various complex applications [14]. The optimization methods developed for nonlinear programming with multiple constraints could be used to solve continuous nonlinear knapsack problems [15]. However, specialized algorithms are much faster and more reliable than standard nonlinear programming software. For solving continuous nonlinear knapsack problems, there are two basic specialized approaches: multiplier search methods (e.g., [16]) and variable pegging methods (e.g., [17]). Multiplier search methods relax the knapsack constraint, and generate iterative solutions until all KKT conditions are satisfied; Pegging methods maintain all KKT conditions during their iterations except the bounds on the variables, till all bounds are satisfied. For a recent literature review on these methods, please refer to [12,13]. Although continuous nonlinear knapsack problems can be solved by applying these two types of methods, as shown in [13], some methods based on the problem structures will be more efficient. The proposed model in this paper is different from the classical nonlinear knapsack problems. By taking advantage of the structural properties of the model, we propose an efficient algorithm for solving the optimal solution. Sensitivity analysis is done for better understanding the behavior of the problem, and numerical results are used to show the performance of the proposed solution method. The paper’s contribution is twofold. First we study the problem with different attributes which are not studied before. Second we propose an efficient algorithm for solving this special knapsack problem. The organization of the remainder of this paper is as follows. Section 2 introduces and models the problem. In Section 3, the structural properties are investigated and the solution method is developed. Section 4 is dedicated to sensitivity analysis. Numerical results are discussed in Section 5, and Section 6 concludes this paper with providing a multi-firm view of the studied problem, and points out some future research directions.
2. Problem formulation Firms often find it profitable to segment customers according to their price sensitivity and to price discriminately accordingly. For examples, retailers of food and beverage face different price elasticity at different locations. As [18] mentioned, in Latin America, McDonalds sells hamburgers for higher prices in wealthy neighborhoods than in poorer ones; a glass of beer costs more at an airport bar than at the corner bar. Research in other industries also found different price elasticity at different locations. Wang [19] showed that gasoline in different stations has different demand curves by estimating station level price elasticity of demand for gasoline in market with regular price cycles; Bernstein and Griffin [20] found that there are regional and state difference in price–demand relationship for electricity and natural gas by examining electricity and natural gas use in the residential sector and electricity use in the commercial sector. Milk products provide a good example of varying price elasticity across submarkets. Veeramani [21] used retail scanner data to show that similar milk products have different demand elasticity and demand curves. Similar patterns were observed by
other researchers. Capps [22] showed elastic demand for halfgallon milk and inelastic demand for gallon milk. Green and Park [23] reported the elasticities of store level weekly milk scanner data as 0.89, 2.19, 1.84, and 2.16 for whole milk, 2% milk, 1% milk and skim milk, respectively. Smed and Jensen [24] found the elasticities for whole milk, reduced fat milk, low fat milk, and skim milk as 2.18, 1.82, 1.32 and 2.25, respectively, by using Danish weekly household panel data. The different products or submarkets in the above examples may share the same capacity or raw materials. For examples, electricity or gasoline sold in different locations may be supplied by the same facility or production factory; different milk products may be produced in the same production line, sharing the same production capacity or raw milk. In automobile industry, different product categories often are assembled in the same assembly line. For example, one Honda automobile factory in China assembles its four brands of cars (Accord, Odyssey, Fit and City) in the unique assembly line although the prices for these cars are varying from $15,000 to $50,000. The characteristics of the above examples can be summarized as follows: (1) different markets have different demand elasticity and demand curves; (2) there are different supply costs of products in different markets, e.g., different milk types (whole milk, 2% milk, 1% milk and skim milk) need different production operations which incurs different costs; (3) there is a common capacity shared in different markets, such as the supply of raw milk or the production capacity of automobile assembly line; (4) there are some market-specific capacity constraints, such as the storage capacities of different gasoline stations, brand-specific component supplies for different brand cars. In this paper, we consider a multiple-market pricing problem with these characteristics. The firm has limited common capacity Q0 to produce the same/similar products, which are sold on n different markets. Each market (i ¼ 1, . . . ,n) has a market-specific capacity Qi, and a market-specific unit cost ci, which may consist of market-specific production cost and market-specific selling cost in market i ¼ 1, . . . ,n. Let pi, and qi(pi) be the price and the demand curve for market i ¼ 1, . . . ,n, respectively. We assume that demand curve qi(pi), i ¼ 1, . . . ,n, is downward sloping, and we also assume d2 qi ðpi Þ=dp2i Z 0 since demand curves are generally convex. These assumptions are often made for demand curves in literature. Let ei ðpi Þ ¼ ðdqi ðpi Þ=dpi Þðpi =qi ðpi ÞÞ be the elasticity derived from qi(pi), i ¼ 1, . . . ,n, we consider two typical demand models, i.e., linear and semi-log linear demands. Table 1 gives some values of these two demand curves. The firm will make the pricing decisions on pi, i ¼ 1, . . . ,n, for maximizing the total profit. Let pi(pi) ¼(pi ci)qi(pi) be the profit obtained from market i ¼ 1, . . . ,n, then the profit maximization problem can be modeled as a continuous nonlinear knapsack problem as follows (denoted as problem P): Max p ¼
n X
pi ðpi Þ ¼
i¼1
n X
ðpi ci Þqi ðpi Þ,
ð1Þ
i¼1
subject to n X
qi ðpi Þ r Q0 ,
ð2Þ
i¼1
Table 1 The values for two demand curves. Demand model
Demand function qi(pi)
Elasticity ei ðpi Þ
pi ð1 þ 1=ei ðpi ÞÞ
Linear Semi-log linear
ai bi pi , ai 4 0, bi 4 0 ai bi logðpi Þ, ai 4 0, bi 4 0
ðbi pi Þ=ðai bi pi Þ bi =ðai bi logðpi ÞÞ
2pi ai =bi pi 1ðai bi logðpi ÞÞ=bi
B. Zhang / Computers & Operations Research 39 (2012) 105–111
0 r qi ðpi Þ rQi , pi Zci ,
i ¼ 1, . . . ,n,
i ¼ 1, . . . ,n:
ð3Þ ð4Þ
In the above problem, constraint (2) specifies the knapsack constraint on the total supply quantity, which reflects that the total sales in all markets should not exceed the shared common capacity Q0. 0rqi(pi) in constraint (3) depicts the non-negative constraint on the selling quantity in each market, and qi(pi)rQi in constraint (3) means the selling quantity in market i should not exceed the market-specific capacity Qi for market i. Constraint (4) indicates the lower bound constraint on pi since the selling price should not be smaller than the market-specific cost. Let pi(qi) be the inverse function of qi(pi), i ¼ 1, . . . ,n. Since demand curve qi(pi), i ¼ 1, . . . ,n, is downward sloping, constraint (3) can be represented as a constraint of pi. 0rqi(pi) in constraint (3) can be rewritten as pi(qi)rui, where ui ¼ argfqi ðpi Þ ¼ 0g. For examples, we have ui ¼ ai =bi for linear demand, and ui ¼ eai =bi for semi-log linear demand. qi(pi)rQi in constraint (3) can be rewritten as pi(qi)Zli, where li ¼ argfqi ðpi Þ ¼ Qi g. For examples, we have li ¼ ðai Qi Þ=bi for linear demand, li ¼ eðai Qi Þ=bi for semi-log linear demand. The monotony of demand curve ensures li rui, i ¼ 1, . . . ,n. Therefore, constraint (3) can be represented as li rpi rui, i ¼ 1, . . . ,n. Combining this equation with constraint (4), we have di rpi r ui ,
i ¼ 1, . . . ,n,
ð5Þ
where di ¼ maxfli ,ci g, i ¼ 1, . . . ,n. Since constraints (3) and (4) can be rewritten as constraint (5), problem P can be represented as maximizing Eq. (1) subject to constraints (2) and (5). Note that problem P is different from the classical nonlinear knapsack problems studied in [13], because knapsack occupation in the classical knapsack problems often increases as the decision variable increases. Since qi(pi), i ¼ 1, . . . ,n, is decreasing in pi, knapsack occupation decreases with the increasing of pi. Thus, problem P cannot be viewed as one direct application of the classical knapsack problems, and it cannot be solved by applying the method proposed in [13]. Some sophisticated numerical methods of equation solving may be modified or extended for solving the studied problem in this paper. Especially, if the demand is linear model, then the studied problem can be modeled as a quadratic programming problem, which can be solved using equation solving method as Shy [7] did or using some traditional quadratic programming solution methods, such as Lemke complementary pivot algorithm [25,26]. However, the market-specific capacity constraint 0 rqi(pi)rQi, i ¼ 1, . . . ,n, will increase sparseness of the iteration matrices in equation solving. The sophisticated numerical methods will be less efficient when performing iterations on sparse matrices for solving large scale problem. In addition, we have not found any efficient algorithm for solving the studied problem with nonlinear demand model. The mixed linear and nonlinear demands will make the optimal decisions more challenging.
3. Properties and algorithm In this section, we first establish the structural properties for the optimal decisions, and then we develop a solution method for problem P. Beginning with the objective function, we have the following proposition. Proposition 1. The profit function p is jointly concave in pi, i ¼ 1, . . . ,n.
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Proof. Downward sloping qi(pi) implies dqi ðpi Þ=dpi Þ r 0, i ¼ 1, . . . ,n. Since ci rpi, and @pi =@pi ¼ qi ðpi Þ þ ðpi ci Þðdqi ðpi Þ=dpi , we have @2 p=@pi pj ¼ 0 for any iaj, i,j ¼ 1, . . . ,n, and @2 pi =@p2i ¼ 2ðdqi ðpi Þ =dpi Þ þ ðpi ci Þðd2 qi ðpi Þ=dp2i Þ, thus we have @2 pi =@p2i ¼ 2bi o0 for linear demand, and @2 pi =@p2i ¼ ðp þ cÞbi =p2 o0 for semi-log linear demand. & This proposition means problem P has unique optimal pricing decision. Denote by RP the relaxation problem of problem P without the knapsack constraint (2), and let p^ i be the optimal solution to problem RP, then the following proposition characterizes the optimal values of p^ i , i ¼ 1, . . . ,n. Proposition 2. The optimal solution to RP p^ i ¼ minfmaxfdi ,p i g,ui g, where p i ¼ argfpi ð1 þ 1=ei ðpi ÞÞ ¼ ci g.
is
Proof. According to Proposition 1, @pi =@pi ¼ qi ðpi Þ þ ðpi ci Þðdqi ðpi Þ=dpi Þ ¼ 0 gives the optimal condition of the unconstrained problem, ci ¼ pi þqi ðpi Þ=ðdqi ðpi Þ=dpi Þ. The definition of elasticity gives qi ðpi Þ=ðdqi ðpi Þ=dpi Þ ¼ pi =ei ðpi Þ. By combining these two equations, we have ci ¼ pi ð1 þ 1=ei ðpi ÞÞ. Let p i ¼ argfpi ð1 þ 1=ei ðpi ÞÞ ¼ ci g, then p i is the optimal solution to the unconstrained problem. Since p is concave, then we have p^ i ¼ minfmaxfdi ,p i g,ui g by directly comparing p i and the bound constraint di rpi rui in constraint (5). & To further analyze the properties of the optimal pricing decisions of problem P, following [27], we define a marginal capacity benefit function of market i ¼ 1, . . . ,n as ri ðpi Þ ¼
dpi =dpi q ðp Þ þ ðpi ci Þdqi ðpi Þ=dpi ¼ i i ¼ pi ð1 þ 1=ei ðpi ÞÞci : dqi ðpi Þ=dpi dqi ðpi Þ=dpi ð4Þ
From the results in Table 1, we know pi ð1 þ1=ei ðpi ÞÞ is increasing in pi, i ¼ 1, . . . ,n, thus ri(pi) is an increasing function of pi, i ¼ 1, . . . ,n. Let pi , i ¼ 1, . . . ,n, be the optimal solution to problem P, and denote by O ¼ fijdi o pi oui ,i ¼ 1, . . . ,ng the set of product indices of the optimal nonbound solutions. Then we have the following proposition. Proposition 3. P (a) If ni¼ 1 qi ðp^ i Þ r Q0 then pi ¼ p^ i , i ¼ 1, . . . ,n. Pn P (b) If i ¼ 1 qi ðp^ i Þ 4 Q0 , then ni¼ 1 qi ðpi Þ ¼ Q0 . (c) rj(pj ) ¼rk(pk ) for all j,k A O. Proof. (1) Property (a) is intuitive since the knapsack constraint is inactive. (2) Property (b) can be proved by using the method of reduction P P to absurdity. If ni¼ 1 qi ðpi Þ oQ0 , according to ni¼ 1 qi ðp^ i Þ 4Q0 , there must exist at least one k A f1, . . . ,ng such that qk(pk ) oqk(p^ k ), which implies that pk 4 p^ k . From dk r p^ k ruk and pk 4 p^ k , we have pk 4dk. According to dk r pk ruk and pk 4 p^ k , we have p^ k o uk . Since p^ k o uk , we have dpk =dp^ k r 0, and hence rk ðp^ k Þ Z 0. Since rk(pk) is increasing in pk, we have rk ðpk Þ 4rk ðp^ k Þ Z 0, which implies dpk =dpk o0. The conditions dpk =dpk o0 and pk 4dk imply that a sufficient small decrease on pk will increase the profit, which violates the optimality of pi , Pn i ¼ 1, . . . ,n. Therefore, there must be i ¼ 1 qi ðpi Þ ¼ Q0 , if Pn ^ q ð p Þ 4 Q . 0 i¼1 i i P n ^ (3) If i ¼ 1 qi ðp i Þ r Q0 , we have ri(pi ) ¼0 for all i A O, and hence property (c) is proved. When the knapsack constraint is binding, property (c) can be proved with reduction to absurdity. If rj(pj ) ark(pk ) for j,kA O, without loss of generality, we assume that rj(pj )ork(pk ). From the proof of property (b), we know dpi =dpi 40 for all i A O, otherwise, a sufficient small decrease on pi will increase the profit. The condition
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dpi =dpi 4 0 implies ri(pi ) o0 for all i A O. Therefore, we have rj(pj ) ork(pj )o0. There exists a sufficient small dk 40 satisfying rj(pj + dj)ork (pk dk) o0 where dj ¼ ððdqk ðpk Þ=dpk Þ=ðdqj ðpj Þ=dpj ÞÞdk at pj ¼pj + dj and pk ¼pk dk. Let p j ¼pj dj, pk ¼ pk + dk, and p ¼ p for all iaj and iak, i ¼ 1, . . . ,n. Since d is a sufficient k i i small positive value, the condition dqk ðp k Þ=dpk dk ¼ dqj ðpj Þ= dp d gives q (p ) q (p )¼q (p ) q (p ), and hence k k j j j j j k k Pnj i ¼ 1 qi ðpi Þ ¼ Q0 . According to the concavity of the profit function, we have pj ðp j Þpj ðpj Þ Zdpj ðpj Þ=dpj dj ¼ rj ðpj Þdqj ðpj Þ=dpj dj , and pk ðp k Þpk ðpk Þ Z dpk ðpk Þ=dpk dk ¼ rk ðpk Þdqk ðpk Þ=dpk dk : Since dqk ðp k Þ=dpk dk ¼ dqj ðpj Þ=dpj dj o 0 and rj(pj ) rk(pk ) o0, combining these two equations gives pj(p ) j pj(pj )+ pk(pk ) pk(pk ) 40. It is in contradiction with the optimality of pi , i ¼ 1, . . . ,n. Thus, we have rj(pj )¼rk(pk ) for all j,k A O. &
Property (a) implies that the optimal pricing decisions of problem P are the same as those of problem RP when the common capacity constraint is not active. Property (b) illustrates that the optimal solution to problem P is obtained when the capacity is fully utilized if the common capacity constraint is active. If there is some remaining common capacity at one solution, we can always increase the total profit by reducing the selling price in some markets to fully utilize the remaining capacity. Property (c) implies that the marginal capacity benefits should be equal for all the optimal nonbound solutions. This property always holds since the total profit can be increased by moving some capacity from the market with smaller marginal capacity benefit to the market with larger marginal capacity benefit. The economic meaning of this property is that the most efficient utilization of resources is achieved at the optimal pricing decisions. Based on the properties of problem P depicted in Proposition 3, we can develop an efficient algorithm for solving problem P. If the knapsack constraint is inactive, then the optimal solution can be solved using Proposition 2; Otherwise, the optimal solution Pn must satisfy i ¼ 1 qi ðpi Þ ¼ Q0 , rj(pj ) ¼rk(pk )¼r* for all j,k A O, which means that it is the key to determine r* for solving the optimal solution. Since ri(pi) is increasing in pi, i ¼ 1, . . . ,n, then r*
Fig. 1. The algorithm for solving problem P.
can be solved by applying a bi-section procedure over r A ½mini ¼ 1,...,n fri ðdi Þg,maxi ¼ 1,...,n fri ðui Þg. Main steps of the algorithm for solving problem P are summarized in Fig. 1. In this algorithm, we first calculate the bound constraints on price variables (Step 0), then we solve problem RP (Step 1) to obtain p^ i . We judge whether p^ i incurs a binding knapsack constraint or not (Step 2). If the knapsack constraint is inactive, the optimal solution is pi ¼ p^ i , i ¼ 1, . . . ,n. Otherwise, we apply a bi-section procedure over interval [rL,rU] to determine pi (i ¼ 1, . . . ,n) (Steps 3–6). Note that pi(r) in Step 4 is the inverse function of ri(pi), and the bound constraints are considered in Step 4. After solving the optimal solution, Steps 7–8 calculate the maximal profit and output the optimal pricing decisions. It is obvious that this algorithm is a polynomial algorithm of the o(n) order.
4. Sensitivity analysis In this section, we investigate how different model parameters affect the optimal pricing decisions. We analyze how the optimal prices change as we change the capacity constraints and the different supply costs of products in different markets. In our sensitivity analysis, if we change one parameter, we keep other parameters unchanged. Based on Propositions 2 and 3, we have the following proposition. Proposition 4. For all i ¼ 1, . . . ,n, (a) (b) (c) (d) (e)
pi is nonincreasing in Q0; pi is nondecreasing in ci; pi is nonincreasing in Qi; pk , kai, k ¼ 1, . . . ,n, is nonincreasing in ci; pk , kai, k ¼ 1, . . . ,n, is nondecreasing in Qi;
Proof. These properties are obvious when the knapsack constraint is inactive. In the following, we prove the case of binding knapsack constraint using the method of reduction to absurdity. Pn (a) If Q0 increases, i ¼ 1 qi ðpi Þ ¼ Q0 in Proposition 3(b) requires that pi for some i ¼ 1, . . . ,n should decrease. If some pi decrease, and others increase, then the equal condition for marginal capacity benefits in Proposition 3(c) will be violated. Thus, pi will never increase when Q0 increase. (b) If ci increases and pi decreases, then ri(pi ) will decrease. In this situation, Proposition 3(c) means the profit can be increased by moving some capacity from market i to other markets, thus decreasing of pi never gives the optimal solution. (c) If Qi increases, then li decreases, and hence di nonincreases. Since di is a lower bound of pi, pi is nonincreasing in Qi; (d) From property (b), we know that pi does not decrease when ci increases. This implies that the selling quantity qi(pi ) in market i will not increase. The remaining capacity saved from the decreasing of qi(pi ) in market i can be utilized in other markets, thus pk , kai, k ¼ 1, . . . ,n will nonincrease. (e) If Qi increases, according to property (c), pi does not increase, and then the selling quantity qi(pi ) in market i will nondecrease, and the required capacity from the increasing of qi(pi ) in market i should be moved from other markets. Thus pk , kai, k ¼ 1, . . . ,n, is nondecreasing in Qi. & From this proposition, we have the following economic insights. Property (a) implies that the prices in all market will not increase and the selling quantities in all markets will not decrease when the shared common capacity increases. Property
B. Zhang / Computers & Operations Research 39 (2012) 105–111
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(b) indicates that if the market-specific cost of one market increases, then the corresponding price (selling quantity) in this market will increase (decrease). Property (c) means that if the market-specific capacity increases, then the corresponding price (selling quantity) in this market will decrease (increase). Property (d) implies that the prices (selling quantities) in other markets will decrease (increase) if the market-specific cost of one market increases. Property (e) indicates the increasing of one marketspecific capacity will result in the increasing (decreasing) of the prices (selling quantities) in other markets.
5. Numerical results In this section, numerical results are provided to illustrate the application of our study in a real-life example, to verify the results of sensitivity analysis and to show the performance of the proposed solution procedure.
Fig. 2. Illustration of urban and rural demand curves.
5.1. Case analysis To illustrate the potential usefulness of the study, we consider a firm that sells refrigerator at average selling price 3400 RMB (100 RMBEUS $15) in Chinese market. Its market is divided into four regional markets (West, North, East, and South) in terms of its operation locations. Each of these four markets is subdivided into two submarkets (urban market and rural market) in light of residents’ income level and logistics costs. Due to the different population densities, the eight submarkets have different potential demands. The elasticity of demand in each submarket is affected by some factors including (1) whether refrigerator is a luxury product or a necessity, and (2) the percentage of income spent on refrigerator. Generally speaking, in urban China, a 3400 RMB refrigerator is viewed as a necessity product, and 3400 RMB is a small amount of money due to residents’ high income; however, in rural China, a 3400 RMB refrigerator is viewed as a luxury product, and 3400 RMB is a large percentage of residents’ low income. Therefore, in rural market, buyers are more responsive to price changes of refrigerator since they view refrigerator as luxury product and will spend a large percentage of their income on it, thus demand in rural markets is more elastic than in urban markets. Since 3400 RMB is a small part of residents’ income in urban market, buyers’ willingness-to-pay around the selling price is uniform, we can model urban demand curves with linear demand model qi(pi)¼ ai bipi with ai 40 and bi 40, i¼1,3,5,7. Since 3400 RMB is a large percentage of residents’ income in rural market, buyers’ willingness-to-pay at 3300 RMB should be larger than that at 3500 RMB, we approximate rural demand curves using semi-log linear demand model, qi ðpi Þ ¼ ai bi logðpi Þ with ai 40 and bi 40, for i¼2,4.6,8. The parameters of the demand models and other parameters used in this example are given in Table 2. The shared common capacity used is Q0 ¼200. Fig. 2 takes q1(p1) and q2(p2) as examples to show urban and rural demand models. Table 2 The parameters of the base case. West
i
ai bi ci Qi
North
East
South
City
Rural
City
Rural
City
Rural
City
Rural
1 160 3.5 15 50
2 140 40 16 30
3 180 4 20 60
4 160 45 21 50
5 200 4 20 50
6 170 48 21 20
7 220 4.5 25 70
8 180 50 26 60
Fig. 3. The values of rL, rU, and r in the iteration process.
To show the efficiency of the proposed algorithm, we plot the iteration process for solving the base case in Fig. 3. In this figure, we report the values of rL, rU, and r in the iteration process. From this figure, it can be observed that within only 22 iteration times, our algorithm terminates at the optimal value r* ¼3.5853 and p* ¼309,242 RMB.
5.2. Sensitivity analysis In this section, we investigate how different model parameters affect the optimal decisions and the optimal profit. In our numerical analysis, if we change one parameter, we keep other parameters unchanged. We present the results of more cases by varying the values of some parameters in the base case. Tables 3–5 provide the results of five cases with different parameters Q0, c8 and Q1, respectively. In these tables, the results of the base case are presented in bold. As shown in Table 3, pi , i ¼ 1, . . . ,8, is decreasing in Q0, which verifies Proposition 4(a), and the optimal profit is increasing in the common capacity. Table 4 shows that p8 increases as c8 increases, but pk , k ¼ 1, . . . ,7, is decreasing in c8, which has been proved in Proposition 4(b) and 4(d) . From Table 4, we also know that the optimal profit will decrease if the selling cost increases. Table 5 confirms Proposition 4(c) and 4(e) since p1 is decreasing in Q1, and pk , k ¼ 2, . . . ,n, is increasing in Q1. We also observe from Table 5 that the optimal profit will increase if the market-specific capacity increases.
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5.3. Statistical results In this subsection, numerical results are provided to show the performance of the proposed algorithm for solving large-scale problems. The instances of problem P are all randomly generated. The two kinds of demand curves are randomly selected in each instance. We use the notation x U(a,b) to denote that x is uniformly generated over [a,b]. The problem parameters are generated as follows: (1) linear demand: ai U(120,200), bi U(2,6), ci U(10,20), Qi U(30,70); (2)semi-log linear demand: ai U(150,200), bi U(30,50), ci U(10,20), Qi U(30,70). In this numerical study, we set n¼50, 100, and 150, and let Q0 ¼n, 2 n, 3 n, and 4 n, respectively. Five levels of ranges are investigated—the given level, twice the ranges, three times the ranges, four times the ranges and five times the ranges of ai, bi, i ¼ 1, . . . ,n. For each size n, Q0, and each level of ranges, 10 test instances are randomly generated. Thus, for each size n, 200 instances with different parameters are solved and evaluated. All Table 3 The optimal solution and profit for varying the capacity Q0. Q0
160 180 200 220 240
p* (100 RMB)
The optimal solution pi (100 RMB) 1
2
3
4
5
6
7
8
33.95 32.92 32.15 31.43 31.43
33.12 31.49 29.18 27.01 24.64
36.09 35.07 34.29 33.56 32.77
35.01 34.32 32.29 30.37 28.28
38.59 37.57 37.50 37.50 37.50
34.52 34.05 32.03 30.14 28.06
40.54 39.51 38.74 38.01 37.21
36.60 36.60 35.37 33.63 31.72
2785.82 2938.81 3092.42 3195.35 3246.08
Table 4 The optimal solution and profit for varying the unit cost c8. c8
11 16 21 26 31
p* (100 RMB)
The optimal solution pi (100 RMB) 1
2
3
4
5
6
7
8
32.46 32.38 32.27 32.15 32.08
30.10 29.86 29.53 29.18 28.96
34.60 34.52 34.41 34.29 34.22
33.10 32.89 32.60 32.29 32.10
37.50 37.50 37.50 37.50 37.50
32.84 32.63 32.34 32.03 31.84
39.05 38.97 38.86 38.74 38.66
30.59 31.73 33.44 35.37 36.60
3203.02 3152.71 3114.52 3092.42 3086.82
Table 5 The optimal solution and profit for varying the upper bound Q1.
p* (100 RMB)
Q1 The optimal solution pi (100 RMB)
10 20 30 40 50
1
2
3
4
5
6
7
8
42.86 40.00 37.14 34.29 32.15
24.64 25.81 27.01 28.24 29.18
32.77 33.16 33.56 33.98 34.29
28.28 29.32 30.37 31.46 32.29
37.50 37.50 37.50 37.50 37.50
28.06 29.09 30.14 31.21 32.03
37.21 37.61 38.01 38.42 38.74
31.72 32.66 33.63 34.61 35.37
2703.22 2906.93 3038.21 3096.62 3092.42
computational experiments are conducted on a laptop computer (dual processor 1.60 GHz, memory 512 M) with Matlab 7.1, and the computation time is reported in milliseconds. The statistical results of number of iterations, computation time, and optimal profits for different size problems are reported in Table 6. In this table, 95% C.I. stands for 95% confidence interval. From Table 6, we come to the conclusion that the proposed algorithm can solve large-scale capacity-constrained price discrimination problems very quickly in limited iterations. Since different ranges for parameters and different capacities are used in our computation, the results in Table 6 can provide a better idea of how well the algorithm behaves. The standard deviations of the optimal profits shown in Table 6 can verify that the ranges of parameters are large. The standard deviations of number of iterations and computation time are quite low, reflecting the fact that the solution method is quite effective and robust. Robustness of the algorithm should be attributed to the efficacy of bi-section procedure, since number of iterations depends strongly on the range of r, which is determined by the values of ai, bi, ci, and Qi. In the bi-section procedure, twice the range of r just requires computation of one more iteration. In addition, the computation time is correlated to the number of iterations times the size of problem, which is verified by the results in Table 6.
6. Conclusions This paper considers a multiple-market price discrimination problem with limited capacities (supplies) in which each market may have different demand elasticity and market-specific capacity, all market share a common capacity. The problem is modeled and solved as a continuous nonlinear knapsack problem via an efficient algorithm. Sensitivity analysis is done for better understanding the behavior of the problem. In our study, we extend the work in literature by considering generalized demand–price elasticity forms, impact of n market case different than 2 market case, and market-specific capacity constraints. Our contribution is twofold. First we consider a more general problem with different attributes. The proposed model simultaneously consider (a) different supply costs of products in different markets, (b) demand in general form including linear and nonlinear forms, (c) multiple-markets, (d) shared common capacity, and (e) market-specific capacity in each market. We provide some structural properties and managerial insights, which are not deeply discussed in literature. Secondly we provide an efficient algorithm for solving the studied problem based on the structural properties of the problem. The proposed algorithm has three main advantages: (1) it can generate exact solution to the studied problem; (2) it has linear computation complexity; (3) it is a unified algorithm for solving the problems with linear, nonlinear or mixed linear and nonlinear demands. Although we use linear, semi-log linear demands as examples in our study, the study can be extended to investigate other cases. In fact, we do not require necessarily
Table 6 Statistical results for randomly generated problems. n
Mean Std. dev. 95% CI Lower Upper
# of iterations
p* (100 RMB)
Computation time
50
100
150
50
100
150
50
100
150
23.78 1.75
24.52 1.63
25.34 1.59
2017.32 417.73
4268.23 622.84
6752.44 946.17
6487.39 2664.34
13,655.04 5438.00
20,257.31 8024.62
23.53 24.02
24.29 24.75
25.11 25.56
1959.07 2075.57
4181.38 4355.08
6620.51 6884.37
6115.88 6858.90
12,896.78 14,413.31
19,138.37 21,376.25
B. Zhang / Computers & Operations Research 39 (2012) 105–111
the difference in demand curves. In this situation, the motivation of price discrimination is to utilize multiple capacities properly instead of dealing with different price elasticity. Our initial objective of this paper is to investigate single-firm multiple-market price discrimination problem. The studied model can also be viewed as a most general model of a market for all firms given that they want to maximize their profits. From this perspective, the shared common capacity should be viewed as the total market demand. Since the objective function of the model is to maximize the total profit of all firms, the model just describes the equilibrium behavior of multiple-firm pricing game with capacity constraints. In the equilibrium situation, the market is the most efficient utilization of resources, according to our equilibrium condition that all firms have the same resource (demand) utilization efficiency (which is reflected by the marginal capacity benefit function r) if their supplies are not tightly restricted by their market-specific capacities (Proposition 3(c)). In this case, any firm will not change its equilibrium price because any change on price will incur price changes from other firms, so it is impossible to make one firm better off without making others worse off for given demand curves. If one firm lower its price in order to get more demand for increasing its profit, other firms will lower their prices to hold their market shares. Since the total demand is limited, the coherent price reduction will result in the demand curves shift to the left, and the total profit of all firms is reduced. This result partly shows that price wars are not good for the companies involved and it can be used to explain why many firms in practice attempt to cooperate for avoiding price wars. Our study presents the equilibrium results of a multi-firm cooperative pricing mechanism. There are different ways to extend this research. One is to investigate the dynamics of the coordination and competition among multiple firms. The proposed model can just be viewed as a static equilibrium model of a market for all firms given that they want to maximize their profits. Understanding the dynamics of the coordination and competition among multiple firms needs further works, including modeling demand substitute, designing competition mechanism, assuming knowledge of competitors, investigating affect of supply–demand, which definitely requires some methodologies (e.g., game theory) different from the ones used in our study. Another area of future research is to investigate the problem integer restrictions on the demand quantities. The corresponding problem can be modeled as a discrete nonlinear programming. Although the analysis on structural properties of the continuous problem in our study can provide some implications for developing heuristics to solve the discrete problem, our initial assessment shows that the discrete problem needs different solution methods. In addition, it would be interesting to examine cases where demand does not decrease when the price increases (e.g. exclusive cars or jewels). The structural properties of the extended problem are significantly different from that of our studied problem. Our initial assessment shows that the study of this problem may require solution methods different from the study in this paper.
Acknowledgements We would like to thank Editor-in-Chief Stefan Nickel and Guest Editors Mhand Hifi and Rym M’Hallah for their work on
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this paper. We really appreciate the reviewers’ insightful comments, which helped to improve the manuscript significantly. This work is supported by national Natural Science Foundation of China (no. 70801065) and the Start Project for Young Teachers at Sun Yat-Sen University.
References [1] Anderson E, Dana Jr JD. When is price discrimination profitable? Northeastern University, working paper, 2008. [2] Malueg DA, Snyder CM. Bounding the relative profitability of price discrimination. International Journal of Industrial Organization 2006;24:995–1011. [3] Reece WS, Sobel SS. Diagrammatic approach to capacity-constrained price discrimination. Southern Economic Journal 2000;66(4):1001–100. [4] Wilson RB. Nonlinear pricing. New York: Oxford University Press; 1993. [5] Baylis K, Perloff JM. Capacity-constrained monopoly. Journal of Industrial Organization Education 2008;3(1):1–13. [6] Bitran G, Caldentey R. An overview of pricing models for revenue management. Manufacturing & Service Operations Management 2003;5(3):203–29. [7] Shy O. How to price: a guide to pricing techniques and yield management. Cambridge, UK: Cambridge University Press; 2008. p. 85–9. [8] Malighetti P, Paleari S, Redondi R. Pricing strategies of low-cost airlines: the Ryanair case study. Journal of Air Transport Management 2009;15:195–203. [9] Dana Jr JD, Yahalom T. Price discrimination with a resource constraint. Economics Letters 2008;100:330–2. [10] Leeson PT, Sobel RS. Costly price discrimination. Economics Letters 2008;99: 206–8. [11] Mujumdar S, Pal D. Do price ceilings abroad increase US drug prices? Economics Letters 2005;87:9–13. [12] Patriksson M. A survey on the continuous nonlinear resource allocation problem. European Journal of Operational Research 2008;185:1–46. [13] Zhang B, Hua Z. A unified method for a class of convex separable nonlinear knapsack problems. European Journal of Operational Research 2008;191: 1–6. [14] Hua Z, Zhang B, Liang L. An approximate dynamic programming approach to convex quadratic knapsack problems. Computers & Operations Research 2006;33(3):660–73. [15] Bretthauer KM, Shetty B. The nonlinear knapsack problem—algorithms and applications. European Journal of Operational Research 2002;138:459–72. [16] Bretthauer KM, Shetty B, Syam S. A branch and bound algorithm for integer quadratic knapsack problems. ORSA Journal on Computing 1995;7: 109–16. [17] Bretthauer KM, Shetty B. A pegging algorithm for the nonlinear resource allocation problem. Computers and Operations Research 2002;29(5):505–27. [18] Phillips R. Pricing and revenue optimization. Stanford, CA: Stanford University Press; 2005. [19] Wang Z. Station level gasoline (petrol) demand in an Australian market with regular price cycles. Northeastern University, working paper, 2008. [20] Bernstein MA, Griffin J. Regional differences in the price-elasticity of demand Energy. Subcontract report, NREL/SR-620-39512, 2006. [21] Veeramani VN. Testing for the existence of price points and price thresholds in retail scanner data using microeconomic theory. PhD thesis, University of Kentucky, 2005. [22] Capps OJ. Demand and marketing analyses for fluid milk products by type and by package size. Paper presented at Dairy Forum, Florida, January 2004. [23] Green GM, Park JL. Retail demand for whole vs. low fat milk: new perspectives on loss leader pricing. Paper presented at the annual American Agricultural Economics Association, Salt Lake City, Utah, August 2–5, 1998. [24] Smed S, Jensen JD. Demand for low fat dairy products—demand for healthiness or taste? Paper presented at the 83rd EAAE seminar, Chania, Crete, September 2002. [25] Lemke CE. Bimatrix equilibrium points and mathematical programming. Management Science 1965;11:681–9. [26] Zhang B, Hua Z. An improved Lemke algorithm for convex quadratic programming with equality constraints. Journal of University of Science and Technology of China 2004;34(6):668–77. [27] Zhang B, Xu X, Hua Z. A binary solution method for the multi-product newsboy problem with budget constraint. International Journal of Production Economics 2009;117:136–41.
Contents lists available at ScienceDirect
Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
Capacity-constrained multiple-market price discrimination Bin Zhang Lingnan College, Sun Yat-Sen University, Guangzhou 510275, PR China
a r t i c l e i n f o
abstract
Available online 5 February 2011
This paper studies a multiple-market price discrimination problem with different markets’ demand elasticity and supply constraints, whereas the markets share a common capacity. We model the problem as a continuous nonlinear knapsack problem, and propose an efficient algorithm for solving the optimal solution after exploring the structural properties of the studied problem. Sensitivity analysis is done for better understanding the behavior of the problem, and numerical results are provided to show the performance of the proposed method. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Price discrimination Nonlinear programming Knapsack problem
1. Introduction Price discrimination has been intensively practiced in sales management when a firm charges different prices for same/ similar products in different markets [1,2]. In sales practice, price discrimination is popular because it often generates more profits over single-price strategy in all markets when the multiple markets have different demand elasticity. Firms in various industries such as airlines, hotels and theaters often face fixed capacity constraints when using price discrimination [3]. Under these circumstances, the shared capacity should be well allocated to different customer groups, and it has important influence on the optimal pricing decisions. The capacity-constrained price discrimination problem has many applications in practice, which has been studied in the last years. Wilson [4] studied deterministic multi-product models by designing an optimal menu of prices and products. Reece and Sobel [3] proposed a diagrammatic approach for solving twomarket case with linear demand model by comparing the marginal costs and marginal revenues. They plotted marginal revenues from two markets and capacity constraint in one figure to find the optimal prices for two markets. This diagrammatic approach cannot be extended for solving multiple-market case, and it cannot deal with market-specific capacity constraints. Baylis and Perloff [5] investigated two-market case through Karush– Kuhn–Tucker (KKT) approach. Bitran and Caldentey [6] modeled the problem as a multiple-product deterministic demand pricing model, and characterized the optimal properties by applying KKT conditions. Shy [7] investigated the problem with linear demand model and without market-specific capacity constraints,
Corresponding author. Tel.: + 86 20 84110649; fax: + 86 20 84114823.
E-mail address: [email protected] 0305-0548/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2011.01.018
and solved the problem by solving a system of equations. Malighetti et al. [8] considered multiple-market pricing problem for low-cost airlines in which the demand for air tickets depends on price levels and on the advanced purchase time, and solved the model through KKT approach. Dana and Yahalom [9] studied price discrimination with a resource constraint by explicitly modeling consumers’ types. In this paper, we extend the research on capacity-constrained price discrimination problem through taking into account the following aspects: (1) we consider the different supply costs of products in different markets, as [10] mentioned, standard theory neglected this costs. (2) We model demand in general forms including linear and nonlinear demand model. (3) We investigate the case of multiple markets instead of two-market case. (4) We explicitly model the shared common capacity as a knapsack constraint. (5) We introduce multiple capacity constraints (one for each market) to reflect other production or market restrictions on selling quantities. Although some of these aspects have been studied in literature, up to now, we have not found any publication that has considered all of them. All these generalizations make the optimal pricing decisions more challenging. For example, Mujumdar and Pal [11] showed that the restriction on decision for one market had important influences on the optimal decision for another market by analyzing a two-market pricing problem. We model the capacity-constrained multiple-market price discrimination as a continuous nonlinear knapsack problem. Continuous nonlinear knapsack problems have been intensively studied in last decades, because of their immediate applications in industry and financial management, but more especially for theoretical reasons. Nonlinear knapsack problems have many applications, such as multi-product newsvendor, production and inventory, Euclidean projection, portfolio selection, video-ondemand batching, and optimum allocation in stratified sampling
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B. Zhang / Computers & Operations Research 39 (2012) 105–111
problems [12,13]. Beside their direct applications, nonlinear knapsack problems have been widely studied because of their frequent appearance as ‘core’ subproblems in various complex applications [14]. The optimization methods developed for nonlinear programming with multiple constraints could be used to solve continuous nonlinear knapsack problems [15]. However, specialized algorithms are much faster and more reliable than standard nonlinear programming software. For solving continuous nonlinear knapsack problems, there are two basic specialized approaches: multiplier search methods (e.g., [16]) and variable pegging methods (e.g., [17]). Multiplier search methods relax the knapsack constraint, and generate iterative solutions until all KKT conditions are satisfied; Pegging methods maintain all KKT conditions during their iterations except the bounds on the variables, till all bounds are satisfied. For a recent literature review on these methods, please refer to [12,13]. Although continuous nonlinear knapsack problems can be solved by applying these two types of methods, as shown in [13], some methods based on the problem structures will be more efficient. The proposed model in this paper is different from the classical nonlinear knapsack problems. By taking advantage of the structural properties of the model, we propose an efficient algorithm for solving the optimal solution. Sensitivity analysis is done for better understanding the behavior of the problem, and numerical results are used to show the performance of the proposed solution method. The paper’s contribution is twofold. First we study the problem with different attributes which are not studied before. Second we propose an efficient algorithm for solving this special knapsack problem. The organization of the remainder of this paper is as follows. Section 2 introduces and models the problem. In Section 3, the structural properties are investigated and the solution method is developed. Section 4 is dedicated to sensitivity analysis. Numerical results are discussed in Section 5, and Section 6 concludes this paper with providing a multi-firm view of the studied problem, and points out some future research directions.
2. Problem formulation Firms often find it profitable to segment customers according to their price sensitivity and to price discriminately accordingly. For examples, retailers of food and beverage face different price elasticity at different locations. As [18] mentioned, in Latin America, McDonalds sells hamburgers for higher prices in wealthy neighborhoods than in poorer ones; a glass of beer costs more at an airport bar than at the corner bar. Research in other industries also found different price elasticity at different locations. Wang [19] showed that gasoline in different stations has different demand curves by estimating station level price elasticity of demand for gasoline in market with regular price cycles; Bernstein and Griffin [20] found that there are regional and state difference in price–demand relationship for electricity and natural gas by examining electricity and natural gas use in the residential sector and electricity use in the commercial sector. Milk products provide a good example of varying price elasticity across submarkets. Veeramani [21] used retail scanner data to show that similar milk products have different demand elasticity and demand curves. Similar patterns were observed by
other researchers. Capps [22] showed elastic demand for halfgallon milk and inelastic demand for gallon milk. Green and Park [23] reported the elasticities of store level weekly milk scanner data as 0.89, 2.19, 1.84, and 2.16 for whole milk, 2% milk, 1% milk and skim milk, respectively. Smed and Jensen [24] found the elasticities for whole milk, reduced fat milk, low fat milk, and skim milk as 2.18, 1.82, 1.32 and 2.25, respectively, by using Danish weekly household panel data. The different products or submarkets in the above examples may share the same capacity or raw materials. For examples, electricity or gasoline sold in different locations may be supplied by the same facility or production factory; different milk products may be produced in the same production line, sharing the same production capacity or raw milk. In automobile industry, different product categories often are assembled in the same assembly line. For example, one Honda automobile factory in China assembles its four brands of cars (Accord, Odyssey, Fit and City) in the unique assembly line although the prices for these cars are varying from $15,000 to $50,000. The characteristics of the above examples can be summarized as follows: (1) different markets have different demand elasticity and demand curves; (2) there are different supply costs of products in different markets, e.g., different milk types (whole milk, 2% milk, 1% milk and skim milk) need different production operations which incurs different costs; (3) there is a common capacity shared in different markets, such as the supply of raw milk or the production capacity of automobile assembly line; (4) there are some market-specific capacity constraints, such as the storage capacities of different gasoline stations, brand-specific component supplies for different brand cars. In this paper, we consider a multiple-market pricing problem with these characteristics. The firm has limited common capacity Q0 to produce the same/similar products, which are sold on n different markets. Each market (i ¼ 1, . . . ,n) has a market-specific capacity Qi, and a market-specific unit cost ci, which may consist of market-specific production cost and market-specific selling cost in market i ¼ 1, . . . ,n. Let pi, and qi(pi) be the price and the demand curve for market i ¼ 1, . . . ,n, respectively. We assume that demand curve qi(pi), i ¼ 1, . . . ,n, is downward sloping, and we also assume d2 qi ðpi Þ=dp2i Z 0 since demand curves are generally convex. These assumptions are often made for demand curves in literature. Let ei ðpi Þ ¼ ðdqi ðpi Þ=dpi Þðpi =qi ðpi ÞÞ be the elasticity derived from qi(pi), i ¼ 1, . . . ,n, we consider two typical demand models, i.e., linear and semi-log linear demands. Table 1 gives some values of these two demand curves. The firm will make the pricing decisions on pi, i ¼ 1, . . . ,n, for maximizing the total profit. Let pi(pi) ¼(pi ci)qi(pi) be the profit obtained from market i ¼ 1, . . . ,n, then the profit maximization problem can be modeled as a continuous nonlinear knapsack problem as follows (denoted as problem P): Max p ¼
n X
pi ðpi Þ ¼
i¼1
n X
ðpi ci Þqi ðpi Þ,
ð1Þ
i¼1
subject to n X
qi ðpi Þ r Q0 ,
ð2Þ
i¼1
Table 1 The values for two demand curves. Demand model
Demand function qi(pi)
Elasticity ei ðpi Þ
pi ð1 þ 1=ei ðpi ÞÞ
Linear Semi-log linear
ai bi pi , ai 4 0, bi 4 0 ai bi logðpi Þ, ai 4 0, bi 4 0
ðbi pi Þ=ðai bi pi Þ bi =ðai bi logðpi ÞÞ
2pi ai =bi pi 1ðai bi logðpi ÞÞ=bi
B. Zhang / Computers & Operations Research 39 (2012) 105–111
0 r qi ðpi Þ rQi , pi Zci ,
i ¼ 1, . . . ,n,
i ¼ 1, . . . ,n:
ð3Þ ð4Þ
In the above problem, constraint (2) specifies the knapsack constraint on the total supply quantity, which reflects that the total sales in all markets should not exceed the shared common capacity Q0. 0rqi(pi) in constraint (3) depicts the non-negative constraint on the selling quantity in each market, and qi(pi)rQi in constraint (3) means the selling quantity in market i should not exceed the market-specific capacity Qi for market i. Constraint (4) indicates the lower bound constraint on pi since the selling price should not be smaller than the market-specific cost. Let pi(qi) be the inverse function of qi(pi), i ¼ 1, . . . ,n. Since demand curve qi(pi), i ¼ 1, . . . ,n, is downward sloping, constraint (3) can be represented as a constraint of pi. 0rqi(pi) in constraint (3) can be rewritten as pi(qi)rui, where ui ¼ argfqi ðpi Þ ¼ 0g. For examples, we have ui ¼ ai =bi for linear demand, and ui ¼ eai =bi for semi-log linear demand. qi(pi)rQi in constraint (3) can be rewritten as pi(qi)Zli, where li ¼ argfqi ðpi Þ ¼ Qi g. For examples, we have li ¼ ðai Qi Þ=bi for linear demand, li ¼ eðai Qi Þ=bi for semi-log linear demand. The monotony of demand curve ensures li rui, i ¼ 1, . . . ,n. Therefore, constraint (3) can be represented as li rpi rui, i ¼ 1, . . . ,n. Combining this equation with constraint (4), we have di rpi r ui ,
i ¼ 1, . . . ,n,
ð5Þ
where di ¼ maxfli ,ci g, i ¼ 1, . . . ,n. Since constraints (3) and (4) can be rewritten as constraint (5), problem P can be represented as maximizing Eq. (1) subject to constraints (2) and (5). Note that problem P is different from the classical nonlinear knapsack problems studied in [13], because knapsack occupation in the classical knapsack problems often increases as the decision variable increases. Since qi(pi), i ¼ 1, . . . ,n, is decreasing in pi, knapsack occupation decreases with the increasing of pi. Thus, problem P cannot be viewed as one direct application of the classical knapsack problems, and it cannot be solved by applying the method proposed in [13]. Some sophisticated numerical methods of equation solving may be modified or extended for solving the studied problem in this paper. Especially, if the demand is linear model, then the studied problem can be modeled as a quadratic programming problem, which can be solved using equation solving method as Shy [7] did or using some traditional quadratic programming solution methods, such as Lemke complementary pivot algorithm [25,26]. However, the market-specific capacity constraint 0 rqi(pi)rQi, i ¼ 1, . . . ,n, will increase sparseness of the iteration matrices in equation solving. The sophisticated numerical methods will be less efficient when performing iterations on sparse matrices for solving large scale problem. In addition, we have not found any efficient algorithm for solving the studied problem with nonlinear demand model. The mixed linear and nonlinear demands will make the optimal decisions more challenging.
3. Properties and algorithm In this section, we first establish the structural properties for the optimal decisions, and then we develop a solution method for problem P. Beginning with the objective function, we have the following proposition. Proposition 1. The profit function p is jointly concave in pi, i ¼ 1, . . . ,n.
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Proof. Downward sloping qi(pi) implies dqi ðpi Þ=dpi Þ r 0, i ¼ 1, . . . ,n. Since ci rpi, and @pi =@pi ¼ qi ðpi Þ þ ðpi ci Þðdqi ðpi Þ=dpi , we have @2 p=@pi pj ¼ 0 for any iaj, i,j ¼ 1, . . . ,n, and @2 pi =@p2i ¼ 2ðdqi ðpi Þ =dpi Þ þ ðpi ci Þðd2 qi ðpi Þ=dp2i Þ, thus we have @2 pi =@p2i ¼ 2bi o0 for linear demand, and @2 pi =@p2i ¼ ðp þ cÞbi =p2 o0 for semi-log linear demand. & This proposition means problem P has unique optimal pricing decision. Denote by RP the relaxation problem of problem P without the knapsack constraint (2), and let p^ i be the optimal solution to problem RP, then the following proposition characterizes the optimal values of p^ i , i ¼ 1, . . . ,n. Proposition 2. The optimal solution to RP p^ i ¼ minfmaxfdi ,p i g,ui g, where p i ¼ argfpi ð1 þ 1=ei ðpi ÞÞ ¼ ci g.
is
Proof. According to Proposition 1, @pi =@pi ¼ qi ðpi Þ þ ðpi ci Þðdqi ðpi Þ=dpi Þ ¼ 0 gives the optimal condition of the unconstrained problem, ci ¼ pi þqi ðpi Þ=ðdqi ðpi Þ=dpi Þ. The definition of elasticity gives qi ðpi Þ=ðdqi ðpi Þ=dpi Þ ¼ pi =ei ðpi Þ. By combining these two equations, we have ci ¼ pi ð1 þ 1=ei ðpi ÞÞ. Let p i ¼ argfpi ð1 þ 1=ei ðpi ÞÞ ¼ ci g, then p i is the optimal solution to the unconstrained problem. Since p is concave, then we have p^ i ¼ minfmaxfdi ,p i g,ui g by directly comparing p i and the bound constraint di rpi rui in constraint (5). & To further analyze the properties of the optimal pricing decisions of problem P, following [27], we define a marginal capacity benefit function of market i ¼ 1, . . . ,n as ri ðpi Þ ¼
dpi =dpi q ðp Þ þ ðpi ci Þdqi ðpi Þ=dpi ¼ i i ¼ pi ð1 þ 1=ei ðpi ÞÞci : dqi ðpi Þ=dpi dqi ðpi Þ=dpi ð4Þ
From the results in Table 1, we know pi ð1 þ1=ei ðpi ÞÞ is increasing in pi, i ¼ 1, . . . ,n, thus ri(pi) is an increasing function of pi, i ¼ 1, . . . ,n. Let pi , i ¼ 1, . . . ,n, be the optimal solution to problem P, and denote by O ¼ fijdi o pi oui ,i ¼ 1, . . . ,ng the set of product indices of the optimal nonbound solutions. Then we have the following proposition. Proposition 3. P (a) If ni¼ 1 qi ðp^ i Þ r Q0 then pi ¼ p^ i , i ¼ 1, . . . ,n. Pn P (b) If i ¼ 1 qi ðp^ i Þ 4 Q0 , then ni¼ 1 qi ðpi Þ ¼ Q0 . (c) rj(pj ) ¼rk(pk ) for all j,k A O. Proof. (1) Property (a) is intuitive since the knapsack constraint is inactive. (2) Property (b) can be proved by using the method of reduction P P to absurdity. If ni¼ 1 qi ðpi Þ oQ0 , according to ni¼ 1 qi ðp^ i Þ 4Q0 , there must exist at least one k A f1, . . . ,ng such that qk(pk ) oqk(p^ k ), which implies that pk 4 p^ k . From dk r p^ k ruk and pk 4 p^ k , we have pk 4dk. According to dk r pk ruk and pk 4 p^ k , we have p^ k o uk . Since p^ k o uk , we have dpk =dp^ k r 0, and hence rk ðp^ k Þ Z 0. Since rk(pk) is increasing in pk, we have rk ðpk Þ 4rk ðp^ k Þ Z 0, which implies dpk =dpk o0. The conditions dpk =dpk o0 and pk 4dk imply that a sufficient small decrease on pk will increase the profit, which violates the optimality of pi , Pn i ¼ 1, . . . ,n. Therefore, there must be i ¼ 1 qi ðpi Þ ¼ Q0 , if Pn ^ q ð p Þ 4 Q . 0 i¼1 i i P n ^ (3) If i ¼ 1 qi ðp i Þ r Q0 , we have ri(pi ) ¼0 for all i A O, and hence property (c) is proved. When the knapsack constraint is binding, property (c) can be proved with reduction to absurdity. If rj(pj ) ark(pk ) for j,kA O, without loss of generality, we assume that rj(pj )ork(pk ). From the proof of property (b), we know dpi =dpi 40 for all i A O, otherwise, a sufficient small decrease on pi will increase the profit. The condition
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dpi =dpi 4 0 implies ri(pi ) o0 for all i A O. Therefore, we have rj(pj ) ork(pj )o0. There exists a sufficient small dk 40 satisfying rj(pj + dj)ork (pk dk) o0 where dj ¼ ððdqk ðpk Þ=dpk Þ=ðdqj ðpj Þ=dpj ÞÞdk at pj ¼pj + dj and pk ¼pk dk. Let p j ¼pj dj, pk ¼ pk + dk, and p ¼ p for all iaj and iak, i ¼ 1, . . . ,n. Since d is a sufficient k i i small positive value, the condition dqk ðp k Þ=dpk dk ¼ dqj ðpj Þ= dp d gives q (p ) q (p )¼q (p ) q (p ), and hence k k j j j j j k k Pnj i ¼ 1 qi ðpi Þ ¼ Q0 . According to the concavity of the profit function, we have pj ðp j Þpj ðpj Þ Zdpj ðpj Þ=dpj dj ¼ rj ðpj Þdqj ðpj Þ=dpj dj , and pk ðp k Þpk ðpk Þ Z dpk ðpk Þ=dpk dk ¼ rk ðpk Þdqk ðpk Þ=dpk dk : Since dqk ðp k Þ=dpk dk ¼ dqj ðpj Þ=dpj dj o 0 and rj(pj ) rk(pk ) o0, combining these two equations gives pj(p ) j pj(pj )+ pk(pk ) pk(pk ) 40. It is in contradiction with the optimality of pi , i ¼ 1, . . . ,n. Thus, we have rj(pj )¼rk(pk ) for all j,k A O. &
Property (a) implies that the optimal pricing decisions of problem P are the same as those of problem RP when the common capacity constraint is not active. Property (b) illustrates that the optimal solution to problem P is obtained when the capacity is fully utilized if the common capacity constraint is active. If there is some remaining common capacity at one solution, we can always increase the total profit by reducing the selling price in some markets to fully utilize the remaining capacity. Property (c) implies that the marginal capacity benefits should be equal for all the optimal nonbound solutions. This property always holds since the total profit can be increased by moving some capacity from the market with smaller marginal capacity benefit to the market with larger marginal capacity benefit. The economic meaning of this property is that the most efficient utilization of resources is achieved at the optimal pricing decisions. Based on the properties of problem P depicted in Proposition 3, we can develop an efficient algorithm for solving problem P. If the knapsack constraint is inactive, then the optimal solution can be solved using Proposition 2; Otherwise, the optimal solution Pn must satisfy i ¼ 1 qi ðpi Þ ¼ Q0 , rj(pj ) ¼rk(pk )¼r* for all j,k A O, which means that it is the key to determine r* for solving the optimal solution. Since ri(pi) is increasing in pi, i ¼ 1, . . . ,n, then r*
Fig. 1. The algorithm for solving problem P.
can be solved by applying a bi-section procedure over r A ½mini ¼ 1,...,n fri ðdi Þg,maxi ¼ 1,...,n fri ðui Þg. Main steps of the algorithm for solving problem P are summarized in Fig. 1. In this algorithm, we first calculate the bound constraints on price variables (Step 0), then we solve problem RP (Step 1) to obtain p^ i . We judge whether p^ i incurs a binding knapsack constraint or not (Step 2). If the knapsack constraint is inactive, the optimal solution is pi ¼ p^ i , i ¼ 1, . . . ,n. Otherwise, we apply a bi-section procedure over interval [rL,rU] to determine pi (i ¼ 1, . . . ,n) (Steps 3–6). Note that pi(r) in Step 4 is the inverse function of ri(pi), and the bound constraints are considered in Step 4. After solving the optimal solution, Steps 7–8 calculate the maximal profit and output the optimal pricing decisions. It is obvious that this algorithm is a polynomial algorithm of the o(n) order.
4. Sensitivity analysis In this section, we investigate how different model parameters affect the optimal pricing decisions. We analyze how the optimal prices change as we change the capacity constraints and the different supply costs of products in different markets. In our sensitivity analysis, if we change one parameter, we keep other parameters unchanged. Based on Propositions 2 and 3, we have the following proposition. Proposition 4. For all i ¼ 1, . . . ,n, (a) (b) (c) (d) (e)
pi is nonincreasing in Q0; pi is nondecreasing in ci; pi is nonincreasing in Qi; pk , kai, k ¼ 1, . . . ,n, is nonincreasing in ci; pk , kai, k ¼ 1, . . . ,n, is nondecreasing in Qi;
Proof. These properties are obvious when the knapsack constraint is inactive. In the following, we prove the case of binding knapsack constraint using the method of reduction to absurdity. Pn (a) If Q0 increases, i ¼ 1 qi ðpi Þ ¼ Q0 in Proposition 3(b) requires that pi for some i ¼ 1, . . . ,n should decrease. If some pi decrease, and others increase, then the equal condition for marginal capacity benefits in Proposition 3(c) will be violated. Thus, pi will never increase when Q0 increase. (b) If ci increases and pi decreases, then ri(pi ) will decrease. In this situation, Proposition 3(c) means the profit can be increased by moving some capacity from market i to other markets, thus decreasing of pi never gives the optimal solution. (c) If Qi increases, then li decreases, and hence di nonincreases. Since di is a lower bound of pi, pi is nonincreasing in Qi; (d) From property (b), we know that pi does not decrease when ci increases. This implies that the selling quantity qi(pi ) in market i will not increase. The remaining capacity saved from the decreasing of qi(pi ) in market i can be utilized in other markets, thus pk , kai, k ¼ 1, . . . ,n will nonincrease. (e) If Qi increases, according to property (c), pi does not increase, and then the selling quantity qi(pi ) in market i will nondecrease, and the required capacity from the increasing of qi(pi ) in market i should be moved from other markets. Thus pk , kai, k ¼ 1, . . . ,n, is nondecreasing in Qi. & From this proposition, we have the following economic insights. Property (a) implies that the prices in all market will not increase and the selling quantities in all markets will not decrease when the shared common capacity increases. Property
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(b) indicates that if the market-specific cost of one market increases, then the corresponding price (selling quantity) in this market will increase (decrease). Property (c) means that if the market-specific capacity increases, then the corresponding price (selling quantity) in this market will decrease (increase). Property (d) implies that the prices (selling quantities) in other markets will decrease (increase) if the market-specific cost of one market increases. Property (e) indicates the increasing of one marketspecific capacity will result in the increasing (decreasing) of the prices (selling quantities) in other markets.
5. Numerical results In this section, numerical results are provided to illustrate the application of our study in a real-life example, to verify the results of sensitivity analysis and to show the performance of the proposed solution procedure.
Fig. 2. Illustration of urban and rural demand curves.
5.1. Case analysis To illustrate the potential usefulness of the study, we consider a firm that sells refrigerator at average selling price 3400 RMB (100 RMBEUS $15) in Chinese market. Its market is divided into four regional markets (West, North, East, and South) in terms of its operation locations. Each of these four markets is subdivided into two submarkets (urban market and rural market) in light of residents’ income level and logistics costs. Due to the different population densities, the eight submarkets have different potential demands. The elasticity of demand in each submarket is affected by some factors including (1) whether refrigerator is a luxury product or a necessity, and (2) the percentage of income spent on refrigerator. Generally speaking, in urban China, a 3400 RMB refrigerator is viewed as a necessity product, and 3400 RMB is a small amount of money due to residents’ high income; however, in rural China, a 3400 RMB refrigerator is viewed as a luxury product, and 3400 RMB is a large percentage of residents’ low income. Therefore, in rural market, buyers are more responsive to price changes of refrigerator since they view refrigerator as luxury product and will spend a large percentage of their income on it, thus demand in rural markets is more elastic than in urban markets. Since 3400 RMB is a small part of residents’ income in urban market, buyers’ willingness-to-pay around the selling price is uniform, we can model urban demand curves with linear demand model qi(pi)¼ ai bipi with ai 40 and bi 40, i¼1,3,5,7. Since 3400 RMB is a large percentage of residents’ income in rural market, buyers’ willingness-to-pay at 3300 RMB should be larger than that at 3500 RMB, we approximate rural demand curves using semi-log linear demand model, qi ðpi Þ ¼ ai bi logðpi Þ with ai 40 and bi 40, for i¼2,4.6,8. The parameters of the demand models and other parameters used in this example are given in Table 2. The shared common capacity used is Q0 ¼200. Fig. 2 takes q1(p1) and q2(p2) as examples to show urban and rural demand models. Table 2 The parameters of the base case. West
i
ai bi ci Qi
North
East
South
City
Rural
City
Rural
City
Rural
City
Rural
1 160 3.5 15 50
2 140 40 16 30
3 180 4 20 60
4 160 45 21 50
5 200 4 20 50
6 170 48 21 20
7 220 4.5 25 70
8 180 50 26 60
Fig. 3. The values of rL, rU, and r in the iteration process.
To show the efficiency of the proposed algorithm, we plot the iteration process for solving the base case in Fig. 3. In this figure, we report the values of rL, rU, and r in the iteration process. From this figure, it can be observed that within only 22 iteration times, our algorithm terminates at the optimal value r* ¼3.5853 and p* ¼309,242 RMB.
5.2. Sensitivity analysis In this section, we investigate how different model parameters affect the optimal decisions and the optimal profit. In our numerical analysis, if we change one parameter, we keep other parameters unchanged. We present the results of more cases by varying the values of some parameters in the base case. Tables 3–5 provide the results of five cases with different parameters Q0, c8 and Q1, respectively. In these tables, the results of the base case are presented in bold. As shown in Table 3, pi , i ¼ 1, . . . ,8, is decreasing in Q0, which verifies Proposition 4(a), and the optimal profit is increasing in the common capacity. Table 4 shows that p8 increases as c8 increases, but pk , k ¼ 1, . . . ,7, is decreasing in c8, which has been proved in Proposition 4(b) and 4(d) . From Table 4, we also know that the optimal profit will decrease if the selling cost increases. Table 5 confirms Proposition 4(c) and 4(e) since p1 is decreasing in Q1, and pk , k ¼ 2, . . . ,n, is increasing in Q1. We also observe from Table 5 that the optimal profit will increase if the market-specific capacity increases.
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5.3. Statistical results In this subsection, numerical results are provided to show the performance of the proposed algorithm for solving large-scale problems. The instances of problem P are all randomly generated. The two kinds of demand curves are randomly selected in each instance. We use the notation x U(a,b) to denote that x is uniformly generated over [a,b]. The problem parameters are generated as follows: (1) linear demand: ai U(120,200), bi U(2,6), ci U(10,20), Qi U(30,70); (2)semi-log linear demand: ai U(150,200), bi U(30,50), ci U(10,20), Qi U(30,70). In this numerical study, we set n¼50, 100, and 150, and let Q0 ¼n, 2 n, 3 n, and 4 n, respectively. Five levels of ranges are investigated—the given level, twice the ranges, three times the ranges, four times the ranges and five times the ranges of ai, bi, i ¼ 1, . . . ,n. For each size n, Q0, and each level of ranges, 10 test instances are randomly generated. Thus, for each size n, 200 instances with different parameters are solved and evaluated. All Table 3 The optimal solution and profit for varying the capacity Q0. Q0
160 180 200 220 240
p* (100 RMB)
The optimal solution pi (100 RMB) 1
2
3
4
5
6
7
8
33.95 32.92 32.15 31.43 31.43
33.12 31.49 29.18 27.01 24.64
36.09 35.07 34.29 33.56 32.77
35.01 34.32 32.29 30.37 28.28
38.59 37.57 37.50 37.50 37.50
34.52 34.05 32.03 30.14 28.06
40.54 39.51 38.74 38.01 37.21
36.60 36.60 35.37 33.63 31.72
2785.82 2938.81 3092.42 3195.35 3246.08
Table 4 The optimal solution and profit for varying the unit cost c8. c8
11 16 21 26 31
p* (100 RMB)
The optimal solution pi (100 RMB) 1
2
3
4
5
6
7
8
32.46 32.38 32.27 32.15 32.08
30.10 29.86 29.53 29.18 28.96
34.60 34.52 34.41 34.29 34.22
33.10 32.89 32.60 32.29 32.10
37.50 37.50 37.50 37.50 37.50
32.84 32.63 32.34 32.03 31.84
39.05 38.97 38.86 38.74 38.66
30.59 31.73 33.44 35.37 36.60
3203.02 3152.71 3114.52 3092.42 3086.82
Table 5 The optimal solution and profit for varying the upper bound Q1.
p* (100 RMB)
Q1 The optimal solution pi (100 RMB)
10 20 30 40 50
1
2
3
4
5
6
7
8
42.86 40.00 37.14 34.29 32.15
24.64 25.81 27.01 28.24 29.18
32.77 33.16 33.56 33.98 34.29
28.28 29.32 30.37 31.46 32.29
37.50 37.50 37.50 37.50 37.50
28.06 29.09 30.14 31.21 32.03
37.21 37.61 38.01 38.42 38.74
31.72 32.66 33.63 34.61 35.37
2703.22 2906.93 3038.21 3096.62 3092.42
computational experiments are conducted on a laptop computer (dual processor 1.60 GHz, memory 512 M) with Matlab 7.1, and the computation time is reported in milliseconds. The statistical results of number of iterations, computation time, and optimal profits for different size problems are reported in Table 6. In this table, 95% C.I. stands for 95% confidence interval. From Table 6, we come to the conclusion that the proposed algorithm can solve large-scale capacity-constrained price discrimination problems very quickly in limited iterations. Since different ranges for parameters and different capacities are used in our computation, the results in Table 6 can provide a better idea of how well the algorithm behaves. The standard deviations of the optimal profits shown in Table 6 can verify that the ranges of parameters are large. The standard deviations of number of iterations and computation time are quite low, reflecting the fact that the solution method is quite effective and robust. Robustness of the algorithm should be attributed to the efficacy of bi-section procedure, since number of iterations depends strongly on the range of r, which is determined by the values of ai, bi, ci, and Qi. In the bi-section procedure, twice the range of r just requires computation of one more iteration. In addition, the computation time is correlated to the number of iterations times the size of problem, which is verified by the results in Table 6.
6. Conclusions This paper considers a multiple-market price discrimination problem with limited capacities (supplies) in which each market may have different demand elasticity and market-specific capacity, all market share a common capacity. The problem is modeled and solved as a continuous nonlinear knapsack problem via an efficient algorithm. Sensitivity analysis is done for better understanding the behavior of the problem. In our study, we extend the work in literature by considering generalized demand–price elasticity forms, impact of n market case different than 2 market case, and market-specific capacity constraints. Our contribution is twofold. First we consider a more general problem with different attributes. The proposed model simultaneously consider (a) different supply costs of products in different markets, (b) demand in general form including linear and nonlinear forms, (c) multiple-markets, (d) shared common capacity, and (e) market-specific capacity in each market. We provide some structural properties and managerial insights, which are not deeply discussed in literature. Secondly we provide an efficient algorithm for solving the studied problem based on the structural properties of the problem. The proposed algorithm has three main advantages: (1) it can generate exact solution to the studied problem; (2) it has linear computation complexity; (3) it is a unified algorithm for solving the problems with linear, nonlinear or mixed linear and nonlinear demands. Although we use linear, semi-log linear demands as examples in our study, the study can be extended to investigate other cases. In fact, we do not require necessarily
Table 6 Statistical results for randomly generated problems. n
Mean Std. dev. 95% CI Lower Upper
# of iterations
p* (100 RMB)
Computation time
50
100
150
50
100
150
50
100
150
23.78 1.75
24.52 1.63
25.34 1.59
2017.32 417.73
4268.23 622.84
6752.44 946.17
6487.39 2664.34
13,655.04 5438.00
20,257.31 8024.62
23.53 24.02
24.29 24.75
25.11 25.56
1959.07 2075.57
4181.38 4355.08
6620.51 6884.37
6115.88 6858.90
12,896.78 14,413.31
19,138.37 21,376.25
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the difference in demand curves. In this situation, the motivation of price discrimination is to utilize multiple capacities properly instead of dealing with different price elasticity. Our initial objective of this paper is to investigate single-firm multiple-market price discrimination problem. The studied model can also be viewed as a most general model of a market for all firms given that they want to maximize their profits. From this perspective, the shared common capacity should be viewed as the total market demand. Since the objective function of the model is to maximize the total profit of all firms, the model just describes the equilibrium behavior of multiple-firm pricing game with capacity constraints. In the equilibrium situation, the market is the most efficient utilization of resources, according to our equilibrium condition that all firms have the same resource (demand) utilization efficiency (which is reflected by the marginal capacity benefit function r) if their supplies are not tightly restricted by their market-specific capacities (Proposition 3(c)). In this case, any firm will not change its equilibrium price because any change on price will incur price changes from other firms, so it is impossible to make one firm better off without making others worse off for given demand curves. If one firm lower its price in order to get more demand for increasing its profit, other firms will lower their prices to hold their market shares. Since the total demand is limited, the coherent price reduction will result in the demand curves shift to the left, and the total profit of all firms is reduced. This result partly shows that price wars are not good for the companies involved and it can be used to explain why many firms in practice attempt to cooperate for avoiding price wars. Our study presents the equilibrium results of a multi-firm cooperative pricing mechanism. There are different ways to extend this research. One is to investigate the dynamics of the coordination and competition among multiple firms. The proposed model can just be viewed as a static equilibrium model of a market for all firms given that they want to maximize their profits. Understanding the dynamics of the coordination and competition among multiple firms needs further works, including modeling demand substitute, designing competition mechanism, assuming knowledge of competitors, investigating affect of supply–demand, which definitely requires some methodologies (e.g., game theory) different from the ones used in our study. Another area of future research is to investigate the problem integer restrictions on the demand quantities. The corresponding problem can be modeled as a discrete nonlinear programming. Although the analysis on structural properties of the continuous problem in our study can provide some implications for developing heuristics to solve the discrete problem, our initial assessment shows that the discrete problem needs different solution methods. In addition, it would be interesting to examine cases where demand does not decrease when the price increases (e.g. exclusive cars or jewels). The structural properties of the extended problem are significantly different from that of our studied problem. Our initial assessment shows that the study of this problem may require solution methods different from the study in this paper.
Acknowledgements We would like to thank Editor-in-Chief Stefan Nickel and Guest Editors Mhand Hifi and Rym M’Hallah for their work on
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this paper. We really appreciate the reviewers’ insightful comments, which helped to improve the manuscript significantly. This work is supported by national Natural Science Foundation of China (no. 70801065) and the Start Project for Young Teachers at Sun Yat-Sen University.
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