OPTIMAL INVESTMENT STRATEGY ON ...

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 12, Number 2, April 2016

doi:10.3934/jimo.2016.12.625 pp. 625–636

OPTIMAL INVESTMENT STRATEGY ON ADVERTISEMENT IN DUOPOLY

Fengjun Wang and Qingling Zhang Institute of Systems Science, Northeastern University Shenyang, Liaoning Province, 110819, China

Bin Li Department of Mathematics and Statistics Curtin University, GPO Box U1987, Perth, WA 6845, Australia

Wanquan Liu Department of Computing, Curtin University, WA, 6102, Australia

(Communicated by Kok Lay Teo) Abstract. In this paper, we will investigate a duopoly competition issue in a commencing period of horizontal expansion. This is an important problem in marketing investment for new products in free market. First, we propose a new market model characterized by nonlinear differential-algebraic equations with continuous inequality constraints, which aims to maximize an enterprise’s product market share rather than its profit in the commencing period in an environment of the duopoly market. In order to solve the investment problem numerically based on proposed model, the control parameterization technique together with the constraint transcription method is used by transforming the proposed problem into a sequence of optimal parameter selection problems. Finally, a practical example on beer sales is used to show the effectiveness of proposed model and we present the optimal advertising strategies corresponding to different competition situations.

1. Introduction. In oligopoly market, there are two types of competitions: price competition and non-price competition. Due to the characteristics of oligopoly market reported in [6], such as small number of enterprises producing the same product, price stability and mutual dependence, the price competition often causes loss to an enterprise. Therefore more and more enterprises adopt the non-price competition strategy in marketing. One effective way for non-price competition is advertising. The Vidale-Wolfe Model reported in [22, 27] is a classical model, which describes the relationship between advertising cost and the change rate of sales. This model can be described by a nonlinear differential equation as follows: x˙ = ρµ(1 − x) − δx, x(0) = x0 ≥ 0,

(1)

2010 Mathematics Subject Classification. Primary: 90-08; Secondary: 49M37. Key words and phrases. Optimal control, nonlinear differential-algebraic model, control parameterization, inequality continuous constraints, investment on advertising. This work was supported by the National Natural Science Foundation of China with Grant No. 61273008 and the National Natural Science Foundation of China with Grant No. 61203001.

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FENGJUN WANG, QINGLING ZHANG, BIN LI AND WANQUAN LIU

where x represents the potential market share of the product, ρ represents advertising efficiency, δ represents recession constant of advertisement and µ represents the advertising cost. There are many results about advertising investment strategies based on this Vidale-Wolfe Model and its variants such as [28, 16, 2, 19] and so on. [18] analyzed each enterprise’s optimal advertising decisions in a duopoly market, where each enterprise’s market share is depending on its own as well as its competitor’s advertising decisions. Such model is extended in [3] where each enterprise considered its advertising policy for a single product. [4] investigated the enterprises’ optimal advertising decisions in an oligopoly market, where each enterprise can sell multiple products. [23] considered the optimal advertising and pricing decisions when an enterprise introduces a new product. [9] extended the dynamic model proposed by [23] to a duopoly setting, where two enterprises compete by making both the advertising and pricing decisions. [20] proposed a sales-advertising response model with a two-dimensional discrete dynamic system, and they further used such model to show that there is a chaos phenomenon in the economic system. [21] also established a dynamic advertising model by considering both of predatory and informative advertising in duopoly. It should be noted that all of these models mentioned above are described by differential equations. And in these models, the initial values of products’ market share are not zero. This is because in these models they are assumed that both enterprises in competition have occupied a certain scale of market shares for their products with certain competitiveness. Also the aim of these models is to maximize the economic profit. In practice, more and more enterprises aim to develop new products for quick potential market occupancy while they have their market with the current products concurrently. For example, Master Kong holds a dominant position in China’s Instant-noodle market, and it also holds a certain percentage in soft drinks market. So the enterprise must make new products win the customers recognition in order to be conducive to the development of new products. In this case, the new product should occupy market share as much as possible in the early development stage. In the process of a new product to be recognized by customers in a free market, the initial value of an enterprise’s product in market share is zero and its goal is to maximize its market share instead of maximizing the profit in the commencing period of horizontal expansion. From zero initial condition, it is unlikely for the enterprise to gain profit before the new product can take some percentage of market shares in competence with other similar products in market. Therefore, it is an interesting and important problem for a new product to achieve the largest market share in a short time period. In comparison with previous researches, the competition goal and the influence factors are also different in this particular case, and this problem has not been investigated in the existing literature. In this paper, we propose a new nonlinear model, which aims to maximize the product market share rather than the economic profit in the commencing period of horizontal expansion. This problem has not been investigated previously. In order to appropriately describe such system, a new differential-algebraic model with continuous inequality constraints is established and investigated. By considering the influence of the advertising, we will define two objective functions with practical significance in this paper. As the derived system is a nonlinear generalized control system. Due to the complexity of the optimal control problem for the nonlinear


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generalized system, it is hard to obtain its analytical solution currently. So we adopt the control parameterizations method reported in [26, 15, 11, 10, 12] and the approach to handle continuous inequality constraints problem reported in [29, 30, 8] to solve this problem numerically. The idea of this method is to partition a finite time interval into several subintervals and the control variables are approximated by piecewise constants in each interval. Then an optimal control problem is approximated by a corresponding optimal parameter selection problem. Along this line we can obtain the numerical solution of the optimal control system. Finally, with a practical example of beer sales, the proposed nonlinear differential-algebraic model is shown to be practicable and effective. And in different competition situations, we can give some corresponding advertising strategies. Through analyzing the experiment results and the affecting factors, some conclusions are given. The organization of this paper is as follows. Section 2 presents a new nonlinear differential-algebraic model describing the starting period in an environment of the oligopoly market with some important aims. Section 3 presents the control parameterizations method. An example on beer sales is demonstrated to show the effectiveness of the proposed model and some results are analyzed in section 4. Section 5 gives the conclusions. 2. Problem Formulation. In this section, a new nonlinear-differential algebraic system is proposed first based on the following analysis. We now consider duopoly enterprises, i.e. enterprise 1 and enterprise 2 selling one same product. In the initial stage, only the enterprise 2’s product is in the current market, and we can assume that the enterprise 2 has gained considerable market share for this product. In this case, the enterprise 1 starts to produce the same product and aims to put the product into the market at some time point. Furthermore, the enterprise 1 is assumed to use the strategy of advertising to obtain the market share. Then we can modify the Vidale-Wolfe Model in (1), and use differential equations with advertising cost as a control variable. At time t, let xi (t) (i = 1, 2) be the state variables which represent the market share of enterprise i’s product and it is calculated by taking the company’s sales over the period and dividing it by the total sales of the industry over the same period. The control variable u(t) ∈ U represents the advertising cost of enterprise 1 where U is the admissible control with U = {u(t)|0 ≤ umin ≤ u(t) ≤ umax }. Clearly, U is compact and convex in this case. Moreover, according to the actual requirement, the market share variable xi (t) should satisfy 0 ≤ xi (t) ≤ 1, so x1 (t) + x2 (t) ≥ 0 is satisfied as well. The dynamic system is expressed as follows x˙ 1 = k1 u(1 − x1 ) − β2 vx1 − η1 x1 (2) x˙ 2 = k2 v(1 − x2 ) − β1 ux2 − η2 x2 where ki > 0 is the influence coefficient of the advertising efficiency, βi > 0 is the advertising competition coefficient, ηi > 0 is the decay rate and v is the advertising cost of enterprise 2. We assume that the enterprise 2 adopts the method of percentage sales reported in [31], i.e., v is changed by the sales of itself and its competitor. Then v(t) is given by v(t) = N (δ1 x1 + δ2 x2 )[a − bN (x1 + x2 )]

(3)

where δi > 0 represents the adjusting parameter of the investment intensity, N > 0 is the total number of customers in market and the formula a − bN (x1 + x2 ) denotes

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the product price which is described by the inverse demand function described in [1], with a and b being the positive constants. Through the above analysis, we can propose a class of generalized systems to represent the product market share competition model:  x˙ 1 = k1 u(1 − x1 ) − β2 vx1 − η1 x1    x˙ 2 = k2 v(1 − x2 ) − β1 ux2 − η2 x2 (4) 0 = N (δ1 x1 + δ2 x2 )[a − bN (x1 + x2 )] − v    x1 (0) = 0, x2 (0) = x20 , v(0) = v0 , t ∈ [0, T ] For convenience, let the system (4) be rewritten as ˙ E Z(t) = f (Z(t), u) where

    1 0 0 x1 E = 0 1 0 , Z = x2  , 0 0 0 v   x˙ 1 = k1 u(1 − x1 ) − β2 vx1 − η1 x1 f (X(t), u) =  x˙ 2 = k2 v(1 − x2 ) − β1 ux2 − η2 x2  N (δ1 x1 + δ2 x2 )[a − bN (x1 + x2 )] − v

(5)

(6)

And the initial condition is written as Z(0) = [0, x20 , v0 ]T . As we analyzed above, there are four continuous inequality constraints, i.e., x1 (t) ≥ 0, 1 − x1 (t) ≥ 0, x2 (t) ≥ 0, 1 − x2 (t) ≥ 0 which are denoted as gi ≥ 0, j = 1, 2, 3, 4. With this model, we aim to answer the following questions. 1) how to design investment strategy on the advertisement such that the enterprise 1 can achieve its market share as much as possible in order to minimize the gap between these two enterprises’ market share in finite time [0, T ]. 2) How to minimize the investment cost to achieve the first aim if possible. 3) check whether it is possible for the enterprise 1 to surpass the enterprise 2 for the market share of product. In summary, we need to solve the following two optimization problems in order to answer these questions academically. (P1 ) minJ1 = x2 (T ) − x1 (T ) u∈U RT (P2 ) minJ2 = x2 (T ) − x1 (T ) + 0 u2 (t)dt u∈U

(P1 ) aims to find an investment strategy with an aim to minimize the product market share in a fixed time interval. (P2 ) aims to solve this problem with minimum cost. These two problems are fundamental in operation research and we aim to solve them in this paper. Based the optimal control theory, Problem (P1 ) may have many solutions as illustrated in an example below and Problem (P2 ) has unique solution in theory. As the model (5) is a nonlinear generalized dynamic system, it is not easy to solve numerically. 3. Constraint Transcriptions and Control Parametrization. Problem (P1 ) and (P2 ) are the nonlinear optimal control problems for the generalized nonlinear systems (5) with continuous inequality constraints. Due to the complexity of the optimal control problem for the nonlinear generalized system, it is difficult to solve them directly by using the optimal control theory for nonlinear generalized system. This is due to a fact that the optimal control problem for nonlinear generalized


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system has not been solved nicely in current literature. In order to solve the proposed problems in last section, we first convert the nonlinear generalized system (5) to a normal nonlinear system. Then we solve the optimal control problem for the corresponding nonlinear dynamic system. However, the resulted optimal control problem is still difficult to be solved since there are rigid requirements to be satisfied at every time point in the time horizon. For optimal control problem of normal nonlinear systems, there are some possible approaches we can use, such as the multiple shooting method in [24] and the direct collocation method in [5]. For the multiple shooting method, an accurate initial guess of the co-state variables is required and furthermore, the optimal solution is very sensitive to this initial guess. The failure of achieving convergence is common for this method. For the direct collocation method, the computational burden can become enormous for effective implementation for large scale problems. In this paper we use the constraint constraint transcription method reported in [25, 15] together with the control parametrization technique in [26, 25, 15, 11, 10, 12] to solve the proposed problems. Once the proposed problems are transformed into a serial of optimal parameter selection problems, then some advanced nonlinear programming methods (For example, the Sequential Quadratic Programming (SQP ) method) can be applied. In addition, the optimal control software package, MISER 3.2 developed by [7], which is implemented based on the control parameterization technique, is available for use. To achieve this goal, Problems (P1 ) and (P2 ) are transformed into optimal control problems in canonical forms by applying the constraint transcription method and then we approximate the transformed problems as a sequence of optimal parameter selection problems by means of the control parametrization technique. Then, the optimal parameter selection problems can be solved as nonlinear mathematical programming problems by applying the gradient based methods. For this, the required gradients for the objective functions and constraint functions will be derived later in this section. To begin, system (5) is converted to an equivalent nonlinear system as below ˙ X(t) = f (X(t), u(t))

(7)

where X=

f (X(t), u(t)) =

x1 , x2

k1 u(1 − x1 ) − β2 x1 N (δ1 x1 + δ2 x2 )[a − bN (x1 + x2 )] − η1 x1 k2 N (1 − x2 )(δ1 x1 + δ2 x2 )[a − bN (x1 + x2 )] − β1 ux2 − η2 x2

The initial condition is written as X(0) = [0, x20 ]T . The continuous inequality constraints are denoted as gj (X(t)) ≥ 0, j = 1, 2, 3, 4. For given system (7) with gj (X(t)) ≥ 0, we can define two problems which are referred to as (P¯1 ) and (P¯2 ) corresponding to the objective functions (P1 ) and (P2 ) respectively. In order to handle the continuous inequality constraints, we shall apply the constraint transcription method as mentioned. First, note that gj (X(t)) ≥ 0 is equivalent to Z T Gj = min{gj (X(t)), 0}dt = 0 (8) 0

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However, (8) is non-differentiable at Gj = 0. To smooth (8) at the ‘sharp corners’, we replace min{gj (X(t)), 0} with Lj,ε (X(t)), where  gj (X(t)) < −ε  gj (X(t)), −(gj (X(t)) − ε)2 /4ε ≥ 0, −ε ≤ gj (X(t)) ≤ ε Lj,ε (X(t)) = (9)  0, gj (X(t)) > ε Thus, for each j = 1, 2, 3, 4,(8) can be expressed approximately as Z T Lj,ε (X(t))dt = 0 Gj,ε =

(10)

0

The equality constraint (10) is differentiable. However, it fails to satisfy the constraint qualification. Thus, we introduce a parameter γ and obtain the following approximate constraint γ + Gj,ε ≥ 0 (11) Remark 1. Obviously, (11) satisfies the constraint qualification by introducing the parameter γ and (11) is in the following canonical form Z T ¯ ¯ Φ(x(T )) + L(x(t), u)dt ≥ 0 (12) 0

¯ ¯ where Φ(x(T )) represents the terminal constraint and L(x(t), u) represents the continuous constraint. By replacing gj (X(t)) ≥ 0 with (11) in Problem (P¯1 ) and Problem (P¯2 ), we can define two related approximate problems, Problem (P¯1ε,γ ) and Problem (P¯2ε,γ ), respectively. The relations between the optimal solutions of Problem (P¯1 ) and Problem (P¯2 ) and the optimal solutions of Problem (P¯1ε,γ ) and Problem (P¯2ε,γ ) are shown by the following lemma. Lemma 3.1. There exists a γ(ε) > 0 such that for all γ, 0 < γ < γ(ε), any feasible control uε,γ of Problem (P¯1ε,γ ) (or Problem (P¯2ε,γ )) is also a feasible control of Problem (P¯1 )(or Problem (P¯2 )). Proof. The proof is similar to the proof for Lemma 8.3.3 reported in [25]. Since Problem (P¯1ε,γ ) and (P¯2ε,γ ) are already in the canonical form. Then, we can apply the control parametrization technique to solve these problems. The idea is shown as follows. We introduce a monotonically non-decreasing sequence {τ0 , τ1 , · · · , τp } which carries out on an equi-partition of [0, T ] and satisfies 0 = τ0 < τ1 < · · · < τp = T . According to the approach reported in [25], the control function u(t) is approximated by a piecewise constant function as follows up (t) =

p X

σk χ[τk−1 ,τk ] (t)

(13)

k=1

where σk ∈ U , χI is the indicator function of I defined by 1, t∈I χI = 0, otherwise Then the system (7) will take the following form ˙ X(t) = fe(X(t), σ p )

(14)


where fe(X(t), σ p ) = f¯(X(t),

p X

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σk χ[τk−1 ,τk ] (t))

k=1

σ p = [σ1 σ2 · · · σp ]T and the initial condition remains the same as X(0). Let Xσp (t) be the solution of p P the system (14) that is generated from the control up (t) = σk χ[τk−1 ,τk ] (t). We k=1

can define Gj (σ p ) =

Z

T

min{gj (Xσp (t)) ≥ 0, 0}dt Z T p Gj,ε (σ ) = Lj,ε (Xσp (t))dt 0

0

And the following system can be obtained  p e ˙  X(t) = f (X(t), σ )  T X(0) = [0, x20 ] p  γ  + Gj,ε (σ ) ≥ 0  p σ ∈ U, t ∈ [0, T ]

(15)

And for the system (15), the optimal control Problem (P¯1ε,γ ) and (P¯2ε,γ ) can be defined, respectively, as follows (P¯ ε,γ (p)) minJ1 (σ p ) = x2,σp (T ) − x1,σp (T ) 1

u∈U

(P¯2ε,γ (p)) minJ2 (σ p ) = x2,σp (T ) − x1,σp (T ) + u∈U (P¯1ε,γ (p))

RT 0

(σ p )2 (t)dt

Problem and Problem (P¯2ε,γ (p)) are a sequence of optimal parameter selection problems in canonical form. As mentioned, the gradient formulas of the objective functions and the constraint functions are needed. We provide these gradient formulas with the following result with the proof omitted since it is similar to the proof of Theorem 5.2.1 reported in [25]. Theorem 3.2. The gradients of the objective functions J1 (σ p ) and J2 (σ p ), and corresponding inequality continuous constraints with respect to σ p are Z T ∂J1 (σ p ) ∂H1 (Xσp (t), σ p , λ1 (t)) = dt (16) ∂σ p ∂σ p 0 Z T ∂J2 (σ p ) ∂H2 (Xσp (t), σ p , λ2 (t)) = dt (17) p ∂σ ∂σ p 0 Z T e ej (t)) ∂Gj,ε (σ p ) ∂ Hj (Xσp (t), σ p , λ = dt (18) ∂σ p ∂σ p 0 e j , j = 1, 2, 3, 4 where H1 and H2 are the Hamiltonian for the objective function, H is the Hamiltonian for the canonical constraints, H1 (Xσp (t), σ p , λ1 (t)) = λT fe(X(t), σ p ) 1

p

2

H2 (Xσp (t), σ , λ2 (t)) = u + λT2 fe(X(t), σ p ) ej (t)) = Lj,ε (Xσp (t)) + λ eT fe(X(t), σ p ) e j (Xσp (t), σ p , λ H j with the boundary conditions ej (T ))T = [0, 0]. (λ1 (T ))T = [−1, 1], (λ2 (T ))T = [−1, 1], (λ

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0.8 x2 States

0.6 0.4 0.2 0

x1 0

1

2

3

4

5

Time

Controls

2 1.5 u

1 0.5 0

0

1

2

3

4

5

Time

Figure 1. The result of system (19) with u(t) = 1, t ∈ [0, 5]

With Theorem 1, Problem (P¯1ε,γ (p)) and Problem (P¯2ε,γ (p)) can be solved easily by the optimal control software package MISER 3.2 developed by [7].

4. An Illustrative Example. In this section, we will use the proposed dynamic system to verify the validity and practicability of the obtained results in this paper. A practical system about beer sales is first established as follows. As we know that all kinds of beer advertisements are a primary means to increase beer sales, and would have impact on consumers’ consumption preferences. Moreover, high advertisement investment usually can achieve higher sales. First, we use the data from [17] for beer advertisement and market share in America during 1993-2003. In detail, all the parameters related to second enterprise in the proposed model are from [17] and the following parameters are used in the proposed model: k1 = 0.5, k2 = 0.5, β1 = 0.5, β2 = 0.7, η1 = 0.1, η2 = 0.2, δ1 = 15, δ2 = 10, a = 1.2 × 10−5 , b = 10−10 , N = 105 . Then the proposed system is as follows.  x˙ 1 = 0.5u(1 − x1 ) − 0.7vx1 − 0.1x1 x˙ 2 = 0.5v(1 − x2 ) − 0.5ux2 − 0.2x2 (19)  v = (15x1 + 10x2 )[1.2 − (x1 + x2 )] where t ∈ [0, 5] and the initial conditions are x1 (0) = 0, x2 (0) = 0.6 and v(0) = 3.6. In general, the beer market share is proportional to the maximum advertising cost umax . Intuitively, if we invest with the maximum expenditure, the product occupancy should be maximized. So we can let u(t) = umax for the whole interval t ∈ [0, T ] in solving Problem (P1 ). Then, we can get the results in Figure 1, and in this case, we have Je = x2 (T ) − x1 (T ) = 0.5151. Next we consider the following two cases.


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States

1 x2 0.5 x1 0

0

1

2

3

4

5

Time

Controls

1

u

0.8 0.6 0.4

0

1

2

3

4

5

Time

Figure 2. The result of optimizing J1 with a better strategy

4.1. The objective function J1 . With above analysis,wee can assume that the optimal result of this problem is J1∗ (u) = 0.5151 as discussed above with the maximal investment cost. In this case, we know that Problem (P1 ) may have many solutions theoretically. Now we are more interested to know whether we can achieve J1∗ (u) = 0.5151 with less investment cost. With such target as a constraint, we can obtain another investment strategy as shown in Figure 2, which shows the information of the competition between two enterprises. Based on the states of system (19) shown in Figure 1 and Figure 2 respectively, we find that they can achieve the same market share at terminal time, but the strategy in Figure 2 is using much lower costs. In view of the actual competition situation in the market, all enterprises are controlling their investment and at same time they expect to achieve the competition goal with the least costs. At the initial time t = 0, the investment costs is usually limited and also x1 (0) = 0. With the growth of the market share, the enterprise 1 can gradually increase advertising. This is in accordance with market rules. Therefore, the advertising strategy in Figure 2 is a better choice for the enterprise 1. 4.2. The objective function J2 with the influence of u. If the enterprise 1 aims to minimize the advertising cost as well as the gap with the second enterprise, we can formulate Problem (P2 ). This is a standard optimal control problem and in this case, we have unique solution in Figure 3. The optimal result of the objective function is J2∗ (u) = 0.8273 in this case. Compared with the result in the previous section, it is clear that the total cost here is reduced but the gap is enlarged as J2∗ > J1∗ . This is reasonable. In above experiments, we found that x2 is always larger than x1 , which implies that the enterprise 1 cannot surpass enterprise 2 in the given time interval. We may ask if it is possible for this to happen. If we solve Problem (P2 ) with umax = 5,

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1 States

x2 0.5

0

0

1

2

3

4

5

x1

Time

Controls

0.4 0.3

u

0.2 0.1 0

0

1

2

3

4

5

Time

Figure 3. The results of optimizing J2

States

1

x1 x2

0.5

0

0

1

2

3

4

5

Time

Controls

5

u

4.5 4 3.5

0

1

2

3

4

5

Time

Figure 4. The result of system (19) with umax = 5 then we can obtain Figure 4. This figure implies that if we increase the investment cost on advertisement, we can make the new product dominant in market. From the above examples, we can find that the proposed model is very useful in controlling the product market share in different cases. Of course, we can derive the minimum investment cost to let the new product be dominant in market share. We will investigate this problem systematically in future.


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5. Conclusions. In this paper, we aim to solve the optimal advertisement cost problem for new product and this is an important problem for investment. For such purpose, we proposed a nonlinear differential-algebraic system with continuous inequality constraints based on the Vidale-Wolfe model for duopoly competition in the oligopoly market first time. Also we formulated some important problems as optimal control problems based on this new model. As to the difficulty for solving these problems in the framework of nonlinear generalized systems, we transform the proposed system into a normal nonlinear dynamic system temporally in this paper and then use the control parametrization technique to solve the proposed problems. In the end, we obtained the numerical solution of such optimal control problems. In the illustrative example, we demonstrated the effectiveness of the proposed model with different investment strategies. In future, we have some new tasks to solve based on observations for results in this paper. First, we need to solve the optimal control problems for the nonlinear generalized system both in theory and numerical way. Such study can enrich nonlinear generalized system theory. Second, we need to design investment strategies based on practical requirement systematically. For example, how to sample the time interval for economical investment? What is the best terminal time, etc? Third, we need to handle the continuous inequality constraints problem by the exact penalty function method mentioned in [14, 13]. REFERENCES [1] B. L. Bai and R. X. Bai, The Modern Western Economic Theory, Economic Science Press, 2011. [2] F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Generic and brand advertising strategies in a dynamic duopoly, Marketing Science, 24 (2005), 556–568. [3] G. M. Erickson, An oligopoly model of dynamic advertising competition, European Journal of Operational Research, 19 (2009), 374–388. [4] G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands, Operations Research, 57 (2009), 1106–1113. [5] G. Fasano and J. Pint´ er, Modeling and Optimization in Space Engineering, Springer, 2013. [6] H. Gao, Western Economics: Macro Part, China Renmin University Press, 2011. [7] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3: Optimal Control Software, Version 2.0, Theory and user manual, 2002. [8] C. H. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30–53. [9] A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durablegood duopoly, European Journal of Operational Research, 200 (2010), 486–497. [10] B. Li, K. L. Teo and G. R. Duan, Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints, International Journal of Innovative Computing, Information and Control, 6 (2010), 3157–3175. [11] B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems–Series B , 16 (2011), 1101–1117. [12] B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications, The ANZIAM Journal, 51 (2009), 162–177. [13] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo’s navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866–875. [14] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260–291.

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