1. Introduction

In the last few decades, the customers are made available with an increased amount of choices for particular goods or services. In such a situation to ensure that the customer chooses our product among others, it becomes important to communicate and inform about the innovative features and quality offered through the product and make a space in customers' minds. This task is achieved by promoting the product at regular intervals. Promotion plays a major role in raising customer awareness of the product, generates sales, and hence repeats purchases. Repeat purchase is an important phenomenon among the consumers that often measures their loyalty towards a brand. The higher is the repeat purchase value, it can be said that the better a firm is doing to keep customers loyal. This chapter focuses on determining the optimal promotional effort policies for a consumer durable product by assuming that the single purchase and the repeat purchase of a product are generated through the combined effect of mass and differentiated promotions in a segmented market.

**14**

*Industrial Engineering*

[1] Ballou RH. Business Logistics

Management. New Jersey: Prentice Hall;

[2] Trent RJ, Handfield RB. Purchasing and Supply Chain Management. Cincinnati, Ohio: South-Western College Pub.; 2nd edition. 2002

[3] Dickson GW. An analysis of vendor selection systems and decisions. Journal

[4] Simpsom PM, Siguaw JA, White SC. Measuring the performance of suppliers: An analysis of evaluation processes. Journal of Supply Chain Management. 2006;**38**(4):29-41

[5] Dempsy WA. Vendor selection and the buying process. Industrial Marketing Management. 1978;**7**:257-267

[6] Weber CA, Current, Benton. Vendor selection criteria and methods. European Journal of Operational

Managerial Auditing Journal.

[7] Askey JM, Dale BG. Internal quality management auditing: An examination.

[8] Boyer K, Verma R. Operations and Supply chain Management for the 21st century. Mason: South-Western Cengage

[9] Lio S, Wiebe H, Enke D. An expert advisory system for the ISO 9001 quality system. Experts Systems with Applications. 2004;**27**(2):313-322

Research. 1991:2-18

1994;**9**(4):3-10

Learning; 2010

of Purchasing. 1966;**1**:5-17

**References**

1999

Promotional strategies are often targeted to a potential market chosen in accordance with the firm's product type. Once target market is decided, market segmentation is carried out to divide the broad target market into subsets of consumers who have common needs and priorities, and then designing and implementing strategies are done to target them. Market segmentation plays an important role in development of the marketing strategies. Different customers have different needs, and it is impossible to satisfy all customers treating them alike. Promotional policies for the products are built by considering the heterogeneity in the potential market. Firms that identify the specific needs of the groups of customers are able to develop the right offer for the submarkets and obtain a competitive advantage over other firms. The concept of market segmentation emerged, as the market-oriented thought evolved among the firms. Market segmentation has thus become the building block of the effective promotional planning. It partitions the markets into groups of potential customers on the basis of geographic, demographic, and psychographic variables and behavioural customer characteristics.

Seidmann et al. [2] proposed a general sales-advertising model in which the state of the system represents a population distribution over a parameter space, and they show that such models are well posed and that there exists an optimal control. Buratto et al. [4] have given some market segmentation concepts into advertising models during the introduction of new product and advertising processes for sales over an infinite horizon. Grosset and Viscolani [3] discussed the optimal advertising policy for a new product introduction considering only the external influence in a segmented market with Nerlove-Arrow's [5] linear goodwill dynamics. Nerlove and Arrow [5] proposed a model in which the effect of advertising on sales is mediated by the goodwill variable. The goodwill state variable represents the effects of the firm investment in advertising, and it affects the demand of the product together with price and other external factors. From past few years, a number of researchers have been working in the area of optimal control models pertaining to advertising expenditure and price in marketing [6]. The simplest diffusion model was due to Bass [7]. Since the landmark work of Bass, the model has been widely used in the diffusion theory. The major limitation of this model is that it does not take into consideration the impact of marketing variables. Many authors have suitably modified the Bass model to study the impact of price on new product diffusion [8–13]. These models incorporate the pricing effects on diffusion. Also there are models that incorporate the effect of advertising on diffusion [9, 14, 15]. Horsky and Simmon [9] incorporated the effects of advertising in the Bass innovation coefficient. Thompson and Teng [16] incorporated learning curve production cost in their oligopoly price-advertising model. Bass et al. [17] included both price and advertis-

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase…

Jha et al. [18] used the concept of market segmentation in diffusion model for advertising a new product and studied the optimal advertising effectiveness rate in a segmented market. They discussed the evolution of sales dynamics in the segmented market under two cases. Firstly, they assumed that the firm advertises in each segment independently, and further they took the case of a single advertising channel, which reaches several segments with a fixed spectrum. Manik et al. [19] amalgamated the two problems formulated by Jha et al. [18] and formulated an optimal control problem where they studied the effect of differentiated promotional effort and mass promotional effort on evolution of sales rate for each segment. They obtained the optimal promotional effort policy for the proposed model. Dynamic behaviour of optimal control theory leads to its application in sales-promotion control analysis and provides a powerful tool for understanding the behaviour of sales-promotion system where dynamic aspect plays an important role. Numerous papers on the application of optimal control theory in sales-advertising problem exist in the literature [20, 21]. However the literature missed out the control model to determine the control policies in a segmented market considering repeat purchasers in the sales through mass and differentiated promotions and taking the

We begin our analysis by stating the following assumption that Mð Þ > 1 is the

number of potential customers of the product in all the segments. The firm simultaneously uses mass market promotion and differentiated market promotion to capture the potential market in each segment, respectively. Mass market promotion reaches each segment proportionally called segment-specific spectrum. Let xið Þt be

<sup>i</sup>¼<sup>1</sup>Xi denotes the total

th segment. During diffusion process,

ing in their generalized Bass model.

DOI: http://dx.doi.org/10.5772/intechopen.81385

budget constraint which we try to do in this chapter.

the number of adopter by time t for the i

total market segments and a discrete variable. The sum ∑<sup>M</sup>

3. Model development

17

Once the segmentation process is complete, the next step following it is choosing the targeting strategies that can be implemented. The firm must decide whether they want to choose segment-specific or mass (differentiated) promotional strategies. Mass promotion is implemented by treating the market as homogeneous and giving common message in all the segments through mass communication, the effect of which reaches each of the segments proportionally known as spectrum effect. However, the preferences of customers may differ, and same offering may not affect all potential customers and urge them towards product adoption. If firms ignore these differences, another competing firm can market similar product serving specific groups, and this may lead to losing customers. Segment-specific promotion recognizes this diversified customer base and takes into consideration the varying consumers in different segments. The promotional messages are constructed accordingly here. Both the mass- and segment-specific strategies play important roles and have their own advantages. Firms generally promote their product in the market at both the levels mass and segment. In this chapter, we assume that the evolution of sales of the product is through mass and differentiated promotions and build a control model for determining the promotional policies that maximizes the total profit constrained on the total budget. The promotion effort policies are generated by using the maximum principle. The model proposed is continuous in nature, but in practical the data available is discrete. Also the model is nonlinear and becomes NP-hard in nature. Thus we have used Lingo11 to solve the discretized version and show the model application.

The rest of this chapter is organized as follows. In Section 2 of this chapter, we provide a brief literature review and in Section 3, we introduce the diffusion model with repeat purchasing and discuss its optimal control formulation and develop segmented sales rate under the assumption that the practitioner may choose independently the advertising intensity directed towards each segment as well as combined advertising intensity. The problem is discussed, and it is solved using Pontryagin's maximum principle with particular cases in Section 4. Section 5 gives the numerical illustration for the discretized version of the problem using Lingo11 software and finally in Section 6, we conclude our chapter.

#### 2. Literature review

Few people have worked in optimal control theory considering market segmentation in advertising models [1–3]. A discrete time stochastic model of multiple media selection in a segmented market was analysed by Little and Lodish [1].

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase… DOI: http://dx.doi.org/10.5772/intechopen.81385

Seidmann et al. [2] proposed a general sales-advertising model in which the state of the system represents a population distribution over a parameter space, and they show that such models are well posed and that there exists an optimal control. Buratto et al. [4] have given some market segmentation concepts into advertising models during the introduction of new product and advertising processes for sales over an infinite horizon. Grosset and Viscolani [3] discussed the optimal advertising policy for a new product introduction considering only the external influence in a segmented market with Nerlove-Arrow's [5] linear goodwill dynamics. Nerlove and Arrow [5] proposed a model in which the effect of advertising on sales is mediated by the goodwill variable. The goodwill state variable represents the effects of the firm investment in advertising, and it affects the demand of the product together with price and other external factors. From past few years, a number of researchers have been working in the area of optimal control models pertaining to advertising expenditure and price in marketing [6]. The simplest diffusion model was due to Bass [7]. Since the landmark work of Bass, the model has been widely used in the diffusion theory. The major limitation of this model is that it does not take into consideration the impact of marketing variables. Many authors have suitably modified the Bass model to study the impact of price on new product diffusion [8–13]. These models incorporate the pricing effects on diffusion. Also there are models that incorporate the effect of advertising on diffusion [9, 14, 15]. Horsky and Simmon [9] incorporated the effects of advertising in the Bass innovation coefficient. Thompson and Teng [16] incorporated learning curve production cost in their oligopoly price-advertising model. Bass et al. [17] included both price and advertising in their generalized Bass model.

Jha et al. [18] used the concept of market segmentation in diffusion model for advertising a new product and studied the optimal advertising effectiveness rate in a segmented market. They discussed the evolution of sales dynamics in the segmented market under two cases. Firstly, they assumed that the firm advertises in each segment independently, and further they took the case of a single advertising channel, which reaches several segments with a fixed spectrum. Manik et al. [19] amalgamated the two problems formulated by Jha et al. [18] and formulated an optimal control problem where they studied the effect of differentiated promotional effort and mass promotional effort on evolution of sales rate for each segment. They obtained the optimal promotional effort policy for the proposed model. Dynamic behaviour of optimal control theory leads to its application in sales-promotion control analysis and provides a powerful tool for understanding the behaviour of sales-promotion system where dynamic aspect plays an important role. Numerous papers on the application of optimal control theory in sales-advertising problem exist in the literature [20, 21]. However the literature missed out the control model to determine the control policies in a segmented market considering repeat purchasers in the sales through mass and differentiated promotions and taking the budget constraint which we try to do in this chapter.

#### 3. Model development

We begin our analysis by stating the following assumption that Mð Þ > 1 is the total market segments and a discrete variable. The sum ∑<sup>M</sup> <sup>i</sup>¼<sup>1</sup>Xi denotes the total number of potential customers of the product in all the segments. The firm simultaneously uses mass market promotion and differentiated market promotion to capture the potential market in each segment, respectively. Mass market promotion reaches each segment proportionally called segment-specific spectrum. Let xið Þt be the number of adopter by time t for the i th segment. During diffusion process,

Promotional strategies are often targeted to a potential market chosen in accordance with the firm's product type. Once target market is decided, market segmentation is carried out to divide the broad target market into subsets of consumers who have common needs and priorities, and then designing and implementing strategies are done to target them. Market segmentation plays an important role in development of the marketing strategies. Different customers have different needs, and it is impossible to satisfy all customers treating them alike. Promotional policies for the products are built by considering the heterogeneity in the potential market. Firms that identify the specific needs of the groups of customers are able to develop the right offer for the submarkets and obtain a competitive advantage over other firms. The concept of market segmentation emerged, as the market-oriented thought evolved among the firms. Market segmentation has thus become the building block of the effective promotional planning. It partitions the markets into groups of potential customers on the basis of geographic, demographic, and psy-

Once the segmentation process is complete, the next step following it is choosing the targeting strategies that can be implemented. The firm must decide whether they want to choose segment-specific or mass (differentiated) promotional strategies. Mass promotion is implemented by treating the market as homogeneous and giving common message in all the segments through mass communication, the effect of which reaches each of the segments proportionally known as spectrum effect. However, the preferences of customers may differ, and same offering may not affect all potential customers and urge them towards product adoption. If firms ignore these differences, another competing firm can market similar product serving specific groups, and this may lead to losing customers. Segment-specific promotion recognizes this diversified customer base and takes into consideration the

chographic variables and behavioural customer characteristics.

varying consumers in different segments. The promotional messages are

discretized version and show the model application.

software and finally in Section 6, we conclude our chapter.

2. Literature review

Industrial Engineering

16

constructed accordingly here. Both the mass- and segment-specific strategies play important roles and have their own advantages. Firms generally promote their product in the market at both the levels mass and segment. In this chapter, we assume that the evolution of sales of the product is through mass and differentiated promotions and build a control model for determining the promotional policies that maximizes the total profit constrained on the total budget. The promotion effort policies are generated by using the maximum principle. The model proposed is continuous in nature, but in practical the data available is discrete. Also the model is nonlinear and becomes NP-hard in nature. Thus we have used Lingo11 to solve the

The rest of this chapter is organized as follows. In Section 2 of this chapter, we provide a brief literature review and in Section 3, we introduce the diffusion model with repeat purchasing and discuss its optimal control formulation and develop segmented sales rate under the assumption that the practitioner may choose independently the advertising intensity directed towards each segment as well as combined advertising intensity. The problem is discussed, and it is solved using Pontryagin's maximum principle with particular cases in Section 4. Section 5 gives the numerical illustration for the discretized version of the problem using Lingo11

Few people have worked in optimal control theory considering market segmentation in advertising models [1–3]. A discrete time stochastic model of multiple media selection in a segmented market was analysed by Little and Lodish [1].

#### Industrial Engineering

repeat purchases of the product may also occur, and those adopters who have already adopted may repurchase the product again. Therefore, the number of adopters for a new product can increase due to both first purchase and repeat purchasing. Under the influence of mass market and differentiated market promotion, evolution of sales rate [7] can be described by the following differential equation:

$$\frac{d\mathbf{x}\_i(t)}{dt} = b\_i(t)(u\_i(t) + a\_i u(t)) \left(\overline{X\_i} - (\mathbf{1} - \mathbf{g}\_i)\mathbf{x}\_i(t)\right), i = \mathbf{1}, \mathbf{2}, \dots, M \tag{1}$$

Max J ¼

dt <sup>¼</sup> pi <sup>þ</sup> qi

4. Solution approach

variable W tðÞ¼ <sup>W</sup><sup>0</sup> � <sup>Ð</sup><sup>t</sup>

Max J ¼

dt <sup>¼</sup> pi <sup>þ</sup> qi

nian can be defined as

0 @ ∑<sup>M</sup>

H ¼

have

19

ðT 0 e �γ<sup>t</sup> ∑<sup>M</sup>

� �

xið Þt Xi

W t \_ ðÞ¼� <sup>∑</sup><sup>M</sup>

lem (5) as

dxið Þt

dxið Þt

ðT 0 e �γ<sup>t</sup> ∑<sup>M</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81385

� �

xið Þt Xi

ables ð Þ uið Þt ; u tð Þ and M state variables ð Þ xið Þt .

<sup>0</sup> <sup>∑</sup><sup>M</sup>

<sup>i</sup>¼1½ðPi � Cið Þ xið Þ<sup>t</sup> x t \_ð Þ� <sup>ϕ</sup>ið Þ uið Þ<sup>t</sup> � � <sup>φ</sup>ð Þ u tð Þ � � dt

� �xið Þ<sup>t</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M

<sup>i</sup>¼1ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � � dt <sup>≤</sup>W<sup>0</sup>

� �xið Þ<sup>t</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M

<sup>i</sup>¼<sup>1</sup>ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � �

xið Þ¼ 0 xi0∀i ¼ 1, 2, …M

<sup>i</sup> ; ui; <sup>u</sup>; <sup>λ</sup>; <sup>μ</sup> � � (8)

xið Þ¼ 0 xi0∀i ¼ 1, 2, …M

9

>>>>>>>>>>>>=

(5)

>>>>>>>>>>>>;

9

>>>>>>>>>>>=

(6)

>>>>>>>>>>>;

A (7)

1

ð Þ uiðÞþt α<sup>i</sup> u tð Þ Xi � 1 � gi

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase…

ðT 0

∑<sup>M</sup>

The above formulated optimal control problem consists of 2M þ 1 control vari-

To solve the above optimal control theory problem, we define a new state

and W Tð Þ≥0. With new state variable, we rewrite the above optimal control prob-

ð Þ uiðÞþt α<sup>i</sup> u tð Þ Xi � 1 � gi

Now, we obtain an optimal control problem with 2M þ 1 control variable and M þ 1 state variable for all segments. Using the maximum principle [22], Hamilto-

<sup>i</sup>¼<sup>1</sup><sup>½</sup> ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup> x t \_ðÞ� <sup>ϕ</sup>ið Þ uið Þ<sup>t</sup> � � <sup>φ</sup>ð Þ u tð Þ

�μð Þ<sup>t</sup> <sup>∑</sup><sup>M</sup>

The Hamiltonian represents the overall profit of the various policy decisions with both the immediate and the future effects taken into account. Assuming the existence of an optimal control solution, the maximum principle provides the necessary optimality conditions; there exist piecewise continuously differentiable functions λið Þt and μð Þt for all t∈½ � 0; T . The value of λið Þt and μð Þt define marginal valuation of state variables xið Þt and W tð Þ at time t, respectively. Here, λið Þt stands for change in future profit as making a small in xið Þt at time t, and μð Þt is the future profit of promotional effort per unit promotion effort expenditure at time t. These variables are known as adjoint variables and describe the similar behaviour in

From the necessary optimality conditions [22, 23] of maximum principle, we

optimal control theory as dual variables in nonlinear programming.

H t; x<sup>∗</sup> <sup>i</sup> ; u<sup>∗</sup> <sup>i</sup> ; u<sup>∗</sup> ; <sup>λ</sup>; <sup>μ</sup> � � <sup>¼</sup> H t; <sup>x</sup><sup>∗</sup>

<sup>i</sup>¼<sup>1</sup>ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � �dt with <sup>W</sup>ð Þ¼ <sup>0</sup> <sup>W</sup><sup>0</sup>

<sup>i</sup>¼<sup>1</sup><sup>½</sup> ð Þ Pi � Cið Þ xið Þ<sup>t</sup> x t \_ðÞ� <sup>ϕ</sup>ið Þ uið Þ<sup>t</sup> � � <sup>φ</sup>ð Þ u tð Þ � � dt

<sup>i</sup>¼<sup>1</sup>ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � �,Wð Þ¼ <sup>0</sup> <sup>W</sup>0,W Tð Þ≥<sup>0</sup>

with the initial condition xið Þ¼ 0 xi0∀i ¼ 1, 2, …, M, where α<sup>i</sup> denotes the segment spectrum of mass promotion α<sup>i</sup> >0& ∑<sup>M</sup> <sup>i</sup>¼1α<sup>i</sup> <sup>¼</sup> <sup>1</sup> � �; gi <sup>0</sup>≤gi <sup>≤</sup><sup>1</sup> � � is susceptible to repeat purchasing, and repeat purchasing is influenced by all factors (both internal and external) affecting first purchase in i th segment by time <sup>t</sup>; uið Þ<sup>t</sup> is differentiated promotional effort rate for i th segment at time <sup>t</sup>; and u tð Þ is mass market promotional effort rate at time t, and bið Þt is the adoption rate per additional adoption for the i th segment. bið Þ<sup>t</sup> can be represented either as a function of time or as a function of the number of previous adopters. Since the latter approach is used most widely, it is the one applied here. Therefore, Eq. (1) can be rewritten as follows:

$$\frac{d\mathbf{x}\_i(t)}{dt} = \left(p\_i + q\_i \frac{\mathbf{x}\_i(t)}{\overline{X\_i}}\right) (u\_i(t) + a\_i \, u(t)) \left(\overline{X\_i} - (\mathbf{1} - \mathbf{g}\_i)\mathbf{x}\_i(t)\right), i = 1, 2, \dots, M \tag{2}$$

where p<sup>i</sup> and q<sup>i</sup> are coefficients of external and internal influences in i th segment, respectively.

The objective of the firm is to maximize the present value of the profit in a planning horizon for a segmented market by selecting optimal mass and differentiated promotional effort rates for the firm. Thus, the objective function can be represented by

$$\mathbf{Max}\,J = \int\_{0}^{T} e^{-\gamma t} \left(\sum\_{i=1}^{M} \left[ (P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t))\dot{\mathbf{x}}(t) - \phi\_i(\mathbf{u}\_i(t))\right] - \rho(\mathbf{u}(t)) \right) dt \tag{3}$$

where ϕið Þ uið Þt and φð Þ u tð Þ are differentiated market promotional effort and mass market promotional effort cost, respectively, γ is discounted profit, Pi is sales price for i th segment, and Cið Þ xið Þ<sup>t</sup> is production cost per unit for <sup>i</sup> th segment, that is, continuous and differentiable with assumption C<sup>0</sup> i ð Þ: > 0 and Pi � Cið Þ xið Þt >0.

During the promotion, differentiated and mass promotions are competing for the limited promotion budget expenditure. Therefore, firms monitor the promotion strategy in all segments closely and allocate their promotional expenditure budget optimally among these segments. The budget constraint for all segments is represented as

$$\int\_{0}^{T} \left(\sum\_{i=1}^{M} \phi\_{i}(u\_{i}(t)) + \rho(u(t))\right) dt \le W\_{0} \tag{4}$$

where W<sup>0</sup> is the fixed budget expenditure for all segments over time. Constraint (4) corresponds to the common promotional expenditure capacity that is allocated among all the segments. This constraint couples the segment and prevents us from simply solving M times a single-segment problem. The above problem can be written as an optimal control problem:

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase… DOI: http://dx.doi.org/10.5772/intechopen.81385

Max J ¼ ðT 0 e �γ<sup>t</sup> ∑<sup>M</sup> <sup>i</sup>¼1½ðPi � Cið Þ xið Þ<sup>t</sup> x t \_ð Þ� <sup>ϕ</sup>ið Þ uið Þ<sup>t</sup> � � <sup>φ</sup>ð Þ u tð Þ � � dt dxið Þt dt <sup>¼</sup> pi <sup>þ</sup> qi xið Þt Xi � �ð Þ uiðÞþt α<sup>i</sup> u tð Þ Xi � 1 � gi � �xið Þ<sup>t</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M xið Þ¼ 0 xi0∀i ¼ 1, 2, …M ðT 0 ∑<sup>M</sup> <sup>i</sup>¼1ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � � dt <sup>≤</sup>W<sup>0</sup> 9 >>>>>>>>>>>>= >>>>>>>>>>>>; (5)

The above formulated optimal control problem consists of 2M þ 1 control variables ð Þ uið Þt ; u tð Þ and M state variables ð Þ xið Þt .

### 4. Solution approach

repeat purchases of the product may also occur, and those adopters who have already adopted may repurchase the product again. Therefore, the number of adopters for a new product can increase due to both first purchase and repeat purchasing. Under the influence of mass market and differentiated market promotion, evolution of sales rate [7] can be described by the following differential

with the initial condition xið Þ¼ 0 xi0∀i ¼ 1, 2, …, M, where α<sup>i</sup> denotes the seg-

ble to repeat purchasing, and repeat purchasing is influenced by all factors (both

market promotional effort rate at time t, and bið Þt is the adoption rate per additional

as a function of the number of previous adopters. Since the latter approach is used most widely, it is the one applied here. Therefore, Eq. (1) can be rewritten as

ð Þ uið Þþt α<sup>i</sup> u tð Þ Xi � 1 � gi

The objective of the firm is to maximize the present value of the profit in a planning horizon for a segmented market by selecting optimal mass and differentiated promotional effort rates for the firm. Thus, the objective function can be

where ϕið Þ uið Þt and φð Þ u tð Þ are differentiated market promotional effort and mass market promotional effort cost, respectively, γ is discounted profit, Pi is sales

During the promotion, differentiated and mass promotions are competing for the limited promotion budget expenditure. Therefore, firms monitor the promotion strategy in all segments closely and allocate their promotional expenditure budget optimally among these segments. The budget constraint for all segments is

> <sup>i</sup>¼<sup>1</sup>ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � �

where W<sup>0</sup> is the fixed budget expenditure for all segments over time. Constraint (4) corresponds to the common promotional expenditure capacity that is allocated among all the segments. This constraint couples the segment and prevents us from simply solving M times a single-segment problem. The above problem can be

th segment, and Cið Þ xið Þ<sup>t</sup> is production cost per unit for <sup>i</sup>

<sup>i</sup>¼<sup>1</sup>½ Þ <sup>ð</sup>Pi � Cið Þ xið Þ<sup>t</sup> x t \_ðÞ� <sup>ϕ</sup>ið Þ uið Þ<sup>t</sup> � � <sup>φ</sup>ð Þ u tð Þ dt

i

where p<sup>i</sup> and q<sup>i</sup> are coefficients of external and internal influences in i

� �

<sup>i</sup>¼1α<sup>i</sup> <sup>¼</sup> <sup>1</sup>

th segment. bið Þ<sup>t</sup> can be represented either as a function of time or

� �xið Þ<sup>t</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M (1)

th segment by time <sup>t</sup>; uið Þ<sup>t</sup> is

th segment at time <sup>t</sup>; and u tð Þ is mass

� �xið Þ<sup>t</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M (2)

; gi <sup>0</sup>≤gi <sup>≤</sup><sup>1</sup> � � is suscepti-

th segment,

(3)

th segment, that

ð Þ: > 0 and Pi � Cið Þ xið Þt >0.

dt ≤W<sup>0</sup> (4)

dt <sup>¼</sup> bið Þ<sup>t</sup> ð Þ uiðÞþ<sup>t</sup> <sup>α</sup>iu tð Þ Xi � <sup>1</sup> � gi

ment spectrum of mass promotion α<sup>i</sup> >0& ∑<sup>M</sup>

internal and external) affecting first purchase in i

differentiated promotional effort rate for i

xið Þt Xi

�

is, continuous and differentiable with assumption C<sup>0</sup>

ðT 0

written as an optimal control problem:

∑<sup>M</sup>

� �

equation:

Industrial Engineering

adoption for the i

follows:

dxið Þt

respectively.

represented by

price for i

represented as

18

Max J ¼

ðT 0 e �γ<sup>t</sup> ∑<sup>M</sup>

dt <sup>¼</sup> pi <sup>þ</sup> qi

dxið Þt

To solve the above optimal control theory problem, we define a new state variable W tðÞ¼ <sup>W</sup><sup>0</sup> � <sup>Ð</sup><sup>t</sup> <sup>0</sup> <sup>∑</sup><sup>M</sup> <sup>i</sup>¼<sup>1</sup>ϕið Þþ uið Þ<sup>t</sup> <sup>φ</sup>ð Þ u tð Þ � �dt with <sup>W</sup>ð Þ¼ <sup>0</sup> <sup>W</sup><sup>0</sup> and W Tð Þ≥0. With new state variable, we rewrite the above optimal control problem (5) as

$$\begin{aligned} \mathbf{Max}\,f &= \int\_0^T e^{-\gamma t} \left( \sum\_{i=1}^M \left( (P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t))) \dot{\mathbf{x}}(t) - \phi\_i(u\_i(t)) \right) - \rho(u(t)) \right) dt \\ \frac{d \mathbf{x}\_i(t)}{dt} &= \left( p\_i + q\_i \frac{\mathbf{x}\_i(t)}{\overline{X\_i}} \right) (u\_i(t) + a\_i \, u(t)) \left( \overline{X\_i} - \left( \mathbf{1} - \mathbf{g}\_i \right) \mathbf{x}\_i(t) \right), i = \mathbf{1, 2, ..., M} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathbf{x}\_i(0) = \mathbf{x}\_{i0} \forall i = \mathbf{1, 2, ...M} \\ \dot{W}(t) &= - \left( \sum\_{i=1}^M \phi\_i(u\_i(t)) + \rho(u(t)) \right), W(0) = W\_0, W(T) \succeq \mathbf{0} \end{aligned} \tag{6}$$

Now, we obtain an optimal control problem with 2M þ 1 control variable and M þ 1 state variable for all segments. Using the maximum principle [22], Hamiltonian can be defined as

$$H = \begin{pmatrix} \sum\_{i=1}^{M} [\left(P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t)) + \lambda\_i(t)\right) \dot{\mathbf{x}}(t) - \phi\_i(u\_i(t))] - \rho(u(t)) \\\\ -\mu(t) \left(\sum\_{i=1}^{M} \phi\_i(u\_i(t)) + \rho(u(t))\right) \end{pmatrix} \tag{7}$$

The Hamiltonian represents the overall profit of the various policy decisions with both the immediate and the future effects taken into account. Assuming the existence of an optimal control solution, the maximum principle provides the necessary optimality conditions; there exist piecewise continuously differentiable functions λið Þt and μð Þt for all t∈½ � 0; T . The value of λið Þt and μð Þt define marginal valuation of state variables xið Þt and W tð Þ at time t, respectively. Here, λið Þt stands for change in future profit as making a small in xið Þt at time t, and μð Þt is the future profit of promotional effort per unit promotion effort expenditure at time t. These variables are known as adjoint variables and describe the similar behaviour in optimal control theory as dual variables in nonlinear programming.

From the necessary optimality conditions [22, 23] of maximum principle, we have

$$H\left(t, \mathfrak{x}\_i^\*, \mathfrak{u}\_i^\*, \mathfrak{u}^\*, \boldsymbol{\lambda}, \boldsymbol{\mu}\right) = H\left(t, \mathfrak{x}\_i^\*, \mathfrak{u}\_i, \mathfrak{u}, \boldsymbol{\lambda}, \boldsymbol{\mu}\right) \tag{8}$$

$$\frac{\partial H^\*}{\partial u\_i} = \mathbf{0} \tag{9}$$

x∗ <sup>i</sup> ðÞ¼ t

x∗ <sup>i</sup> ðÞ¼ t

dλið Þt

Xi

qi Xi

pi þqi xið Þ 0 Xi Xi� <sup>1</sup>�<sup>g</sup> ð Þ<sup>i</sup> xið Þ <sup>0</sup> !

DOI: http://dx.doi.org/10.5772/intechopen.81385

þqi xið Þ 0 Xi Xi� <sup>1</sup>�<sup>g</sup> ð Þ<sup>i</sup> xið Þ <sup>0</sup> !

If xið Þ¼ 0 0, then we get the following result:

1 � exp � qi þ pi 1 � gi

dt <sup>¼</sup> γ λið Þ�<sup>t</sup> ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup>

cost of promotional effort per unit time towards i

pi

and adjoint trajectory is given as

profit of having one more unit of sales is

�γt ðT t e

costs are linear functions

∑<sup>M</sup>

0 @

þ 1 � gi � � pi

1 � gi � � <sup>þ</sup> qi

λiðÞ¼ t e

4.1 Particular cases

can be defined as

H ¼

principle are given by

21

exp qi þ pi 1 � gi � � � � Ð<sup>t</sup>

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase…

� � � � Ð<sup>t</sup>

�γ<sup>s</sup> ð Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup>

� � � � Ð<sup>t</sup>

exp � qi þ pi 1 � gi

<sup>i</sup> ð Þþ <sup>τ</sup> <sup>α</sup>iu<sup>∗</sup>ð Þ<sup>τ</sup> � �d<sup>τ</sup> � � ! � pi

exp qi þ pi 1 � gi � � � � Ð<sup>t</sup>

<sup>0</sup> u<sup>∗</sup> <sup>i</sup> ð Þþ <sup>τ</sup> <sup>α</sup>iu<sup>∗</sup>ð Þ<sup>τ</sup> � �d<sup>τ</sup> � �

<sup>0</sup> u<sup>∗</sup>

∂x\_i ∂xi � �

∂x\_i ∂xi � �

with transversality condition λið Þ¼ T 0. Integrating (20), the value of future

4.1.1 When differentiated market promotional effort and mass market promotional effort

Let us assume that differentiated market promotional effort and mass market promotional effort costs take the following linear forms: ϕið Þ¼ uið Þt κ<sup>i</sup> uið Þt and φð Þ¼ u tð Þ κ u tð Þ and ai ≤uið Þt ≤ Ai, a≤ u tð Þ≤ A, where ai, Ai, a, and A are positive constants which are minimum and maximum acceptable promotional effort rates (ai, Ai, a, and A are determined by the promotional budget) and κ<sup>i</sup> is the per unit

cost of promotional effort per unit time towards mass market. Now, Hamiltonian

<sup>i</sup>¼<sup>1</sup> ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup> <sup>x</sup>\_ <sup>½</sup> <sup>i</sup>ðÞ�<sup>t</sup> <sup>κ</sup><sup>i</sup> uið Þ<sup>t</sup> � � <sup>κ</sup> u tð Þ

�μð Þ<sup>t</sup> <sup>∑</sup><sup>M</sup>

Since Hamiltonian is linear in uið Þt and u tð Þ, optimal differentiated market promotional effort and mass market promotional effort as obtained by the maximum

> ðÞ¼ <sup>t</sup> a if D <sup>≤</sup><sup>0</sup> A if D > 0 �

ai if Bi ≤0 Ai if Bi > 0 �

u∗ <sup>i</sup> ðÞ¼ t

u∗

<sup>i</sup>¼<sup>1</sup>κ<sup>i</sup> uiðÞþ<sup>t</sup> <sup>κ</sup> u tð Þ � �

<sup>i</sup> ð Þþ <sup>τ</sup> <sup>α</sup>iu<sup>∗</sup>ð Þ<sup>τ</sup> � �d<sup>τ</sup> � � , i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M

� x\_ið Þt

� x\_ <sup>i</sup>ð Þt

� � � � dt (21)

∂Ci ∂xi

� � � � (20)

<sup>∂</sup>Cið Þ xið Þ<sup>t</sup> <sup>∂</sup>xið Þ<sup>t</sup>

th segment and κ is the per unit

1

A (22)

(23)

(24)

<sup>0</sup> u<sup>∗</sup>

<sup>0</sup> u<sup>∗</sup> <sup>i</sup> ð Þþ <sup>τ</sup> <sup>α</sup>iu<sup>∗</sup>ð Þ<sup>τ</sup> � �d<sup>τ</sup> � � (18)

(19)

$$\frac{\partial H^\*}{\partial u} = 0 \tag{10}$$

$$\frac{d\lambda\_i(t)}{dt} = \chi \,\lambda\_i(t) - \frac{\partial H^\*}{\partial \mathbf{x}\_i(t)}, \lambda\_i(T) = \mathbf{0} \tag{11}$$

$$\frac{d\mu(t)}{dt} = \lambda\,\mu(t) - \frac{\partial H^\*}{\partial W(t)}, \mu(T) \succeq 0,\tag{12}$$

$$\mathcal{W}(T) + \mathcal{W}\_0 \succeq 0,\\ \mu(T)(\mathcal{W}(T) + \mathcal{W}\_0) = \mathbf{0} \tag{13}$$

Here, μð Þ T ≥0,W Tð Þþ W0≥0, μð Þ T ðW Tð Þþ W0Þ ¼ 0 are called as transversality conditions for W tð Þ. Here, Hamiltonian is independent to W tð Þ, and then we have <sup>μ</sup>\_ <sup>¼</sup> γμ � <sup>∂</sup><sup>H</sup> <sup>∂</sup><sup>W</sup> ¼)μðÞ¼ <sup>t</sup> <sup>μ</sup><sup>T</sup> <sup>e</sup>γð Þ <sup>t</sup>�<sup>T</sup> . Hence, it is clear that the multiplier associated with any integral constraint is constant over time irrespective of their nature (i.e. whether equality or inequality). The Hamiltonian H of each of the segments is strictly concave in uið Þt and u tð Þ. According to the Mangasarian sufficiency theorem [22, 23], there exist unique values of promotional effort controls u<sup>∗</sup> <sup>i</sup> ð Þ<sup>t</sup> and <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> for each segment, respectively. From Eqs. (9) and (10), we get

$$u\_i^\*(t) = \phi\_i^{-1} \left( \frac{(P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t)) + \lambda\_i(t)) \frac{\dot{\mathbf{d}} \dot{\mathbf{x}}\_i(t)}{\mathbf{d}u\_i} - \frac{\partial \mathbf{C}\_i}{\partial \mathbf{x}\_i} \frac{\dot{\mathbf{d}} \dot{\mathbf{x}}\_i}{\partial \dot{u}\_i} \dot{\mathbf{x}}\_i}{\mathbf{1} + \mu\_T \,\, \sigma^{r(t-T)}} \right), i = \mathbf{1, 2, ..., M} \qquad \text{(14)}$$

$$u(t) = \boldsymbol{\rho}^{-1} \left( \frac{\sum\_{i=1}^M \left( (P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t)) + \lambda\_i(t)) \frac{\dot{\mathbf{d}} \dot{\mathbf{x}}\_i(t)}{\mathbf{d}u\_i} \right) - \frac{\partial \mathbf{C}\_i}{\partial \mathbf{x}\_i} \frac{\dot{\mathbf{d}} \dot{\mathbf{x}}\_i}{\mathbf{d}u\_i} \dot{\mathbf{x}}\_i}{\mathbf{1} + \mu\_T \,\, \sigma^{r(t-T)}} \right), i = \mathbf{1, 2, ..., M} \quad \text{(15)}$$

where ϕ�<sup>1</sup> <sup>i</sup> and φ�<sup>1</sup> are inverse functions of ϕ<sup>i</sup> and φ, respectively. If we assume product cost is independent to xið Þt , i.e. Cið Þ¼ xið Þt Ci, then optimal promotional effort policies for each segment become

$$u\_i^\*(t) = \phi\_i^{-1}\left(\frac{(P\_i - \mathbf{C}\_i + \lambda\_i(t))\left(p\_i + q\_i \frac{\mathbf{x}\_i(t)}{\overline{X\_i}}\right)(\overline{X\_i} - (\mathbf{1} - \mathbf{g}\_i)\mathbf{x}\_i(t))}{\mathbf{1} + \mu\_T \, e^{r(t-T)}}\right), i = \mathbf{1}, 2, \dots, M\tag{16}$$

$$u(t) = \rho^{-1}\left(\frac{\sum\_{i=1}^{M}\left((P\_i - C\_i + \lambda\_i(t))\right)a\_i\left(p\_i + q\_i \frac{\mathbf{x}\_i(t)}{X\_i}\right)\overline{(X\_i - (1 - g\_i)\mathbf{x}\_i(t))}\right)}{1 + \mu\_T \,\mathrm{e}^{r(t-T)}}\right), i = 1, 2, \ldots, M\tag{17}$$

The optimal control promotional policy shows that when market is almost saturated, then differentiated market promotional expenditure rate and mass market promotional expenditure rate, respectively, should be zero (i.e. there is no need of promotion in the market).

For optimal control policy, the optimal sales trajectory using optimal values of differentiated market promotional effort u<sup>∗</sup> <sup>i</sup> ð Þ<sup>t</sup> � � and mass market promotional effort <sup>u</sup><sup>∗</sup> ð Þ ð Þ<sup>t</sup> rates for each segment are given by

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase… DOI: http://dx.doi.org/10.5772/intechopen.81385

$$\begin{split} \boldsymbol{x}\_{i}^{\*}(t) &= \frac{\overline{\boldsymbol{X}\_{i}}\left( \left( \frac{\boldsymbol{p}\_{i} + \boldsymbol{q}\_{i}\frac{\boldsymbol{x}\_{i}(0)}{\boldsymbol{X}\_{i}}}{\overline{\boldsymbol{X}\_{i}} - (\boldsymbol{1} - \boldsymbol{g}\_{i})\boldsymbol{x}\_{i}(0)} \right) \exp\left( (\boldsymbol{q}\_{i} + \boldsymbol{p}\_{i}(\boldsymbol{1} - \boldsymbol{g}\_{i})) \int\_{0}^{t} (\boldsymbol{u}\_{i}^{\*}(\boldsymbol{\tau}) + \boldsymbol{a}\_{i}\boldsymbol{u}^{\*}(\boldsymbol{\tau})) d\boldsymbol{\tau} \right) \right) - \boldsymbol{p}\_{i}}{\frac{q\_{i}}{\underline{\boldsymbol{X}\_{i}}} + \left( \mathbf{1} - \boldsymbol{g}\_{i} \right) \left( \frac{p\_{i} + \boldsymbol{q}\_{i}\frac{\boldsymbol{x}\_{i}(0)}{\boldsymbol{X}\_{i}}}{\overline{\boldsymbol{X}\_{i}} - (\boldsymbol{1} - \boldsymbol{g}\_{i})\boldsymbol{x}\_{i}(0)} \right) \exp\left( (\boldsymbol{q}\_{i} + \boldsymbol{p}\_{i}(\boldsymbol{1} - \boldsymbol{g}\_{i})) \int\_{0}^{t} (\boldsymbol{u}\_{i}^{\*}(\boldsymbol{\tau}) + \boldsymbol{a}\_{i}\boldsymbol{u}^{\*}(\boldsymbol{\tau})) d\boldsymbol{\tau} \right) \end{split} \tag{18}$$

If xið Þ¼ 0 0, then we get the following result:

$$\mathbf{x}\_{i}^{\*}(t) = \frac{\mathbf{1} - \exp\left(-\left(q\_{i} + p\_{i}\left(\mathbf{1} - \mathbf{g}\_{i}\right)\right) \int\_{0}^{t} \left(u\_{i}^{\*}(\tau) + a\_{i}u^{\*}(\tau)\right)d\tau\right)}{\left(\mathbf{1} - \mathbf{g}\_{i}\right) + \frac{q\_{i}}{p\_{i}} \left[-\left(q\_{i} + p\_{i}\left(\mathbf{1} - \mathbf{g}\_{i}\right)\right) \int\_{0}^{t} \left(u\_{i}^{\*}(\tau) + a\_{i}u^{\*}(\tau)\right)d\tau\right]}, i = \mathbf{1}, \mathbf{2}, \dots, \mathbf{M} \tag{19}$$

and adjoint trajectory is given as

$$\frac{d\lambda\_i(t)}{dt} = \chi \,\lambda\_i(t) - \left\{ \left( P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t)) + \lambda\_i(t) \right) \left( \frac{\partial \dot{\mathbf{x}}\_i}{\partial \mathbf{x}\_i} \right) - \dot{\mathbf{x}}\_i(t) \left( \frac{\partial \mathbf{C}\_i(\mathbf{x}\_i(t))}{\partial \mathbf{x}\_i(t)} \right) \right\} \tag{20}$$

with transversality condition λið Þ¼ T 0. Integrating (20), the value of future profit of having one more unit of sales is

$$\lambda\_i(t) = e^{-\gamma t} \int\_t^T e^{-\gamma t} \left( (P\_i - \mathbf{C}\_i + \lambda\_i(t)) \left( \frac{\partial \dot{\mathbf{x}}\_i}{\partial \mathbf{x}\_i} \right) - \dot{\mathbf{x}}\_i(t) \left( \frac{\partial \mathbf{C}\_i}{\partial \mathbf{x}\_i} \right) \right) dt \tag{21}$$

#### 4.1 Particular cases

∂H<sup>∗</sup> ∂ui

∂H<sup>∗</sup>

∂H<sup>∗</sup>

∂H<sup>∗</sup>

W Tð Þþ W0≥0, μð Þ T ðW Tð Þþ W0Þ ¼ 0 (13)

<sup>∂</sup><sup>W</sup> ¼)μðÞ¼ <sup>t</sup> <sup>μ</sup><sup>T</sup> <sup>e</sup>γð Þ <sup>t</sup>�<sup>T</sup> . Hence, it is clear that the

1

1

A, i ¼ 1, 2, …, M (14)

A, i ¼ 1, 2, …, M (15)

A, i ¼ 1, 2, …, M

(16)

(17)

A, i ¼ 1, 2, …, M

1

1

dλið Þt

dμð Þt

and then we have <sup>μ</sup>\_ <sup>¼</sup> γμ � <sup>∂</sup><sup>H</sup>

0 @

effort policies for each segment become

0 @

effort controls u<sup>∗</sup>

Industrial Engineering

and (10), we get

<sup>i</sup> ðÞ¼ <sup>t</sup> <sup>ϕ</sup>�<sup>1</sup> i

u tðÞ¼ <sup>φ</sup>�<sup>1</sup> <sup>∑</sup><sup>M</sup>

where ϕ�<sup>1</sup>

<sup>i</sup> ðÞ¼ <sup>t</sup> <sup>ϕ</sup>�<sup>1</sup> i

u tðÞ¼ <sup>φ</sup>�<sup>1</sup> <sup>∑</sup><sup>M</sup>

0 @

of promotion in the market).

differentiated market promotional effort u<sup>∗</sup>

effort <sup>u</sup><sup>∗</sup> ð Þ ð Þ<sup>t</sup> rates for each segment are given by

u∗

20

0 @

u∗

dt <sup>¼</sup> γ λiðÞ�<sup>t</sup>

dt <sup>¼</sup> λ μðÞ�<sup>t</sup>

Here, μð Þ T ≥0,W Tð Þþ W0≥0, μð Þ T ðW Tð Þþ W0Þ ¼ 0 are called as transversality conditions for W tð Þ. Here, Hamiltonian is independent to W tð Þ,

multiplier associated with any integral constraint is constant over time

ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup> \_ <sup>∂</sup>xið Þ<sup>t</sup>

<sup>i</sup>¼<sup>1</sup> ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup> \_ <sup>∂</sup>xið Þ<sup>t</sup>

ð Þ Pi � Ci þ λið Þt pi þ qi

<sup>i</sup>¼<sup>1</sup> ð Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup> <sup>α</sup><sup>i</sup> pi <sup>þ</sup> qi

� �

1 þ μ<sup>T</sup> eγð Þ <sup>t</sup>�<sup>T</sup>

1 þ μ<sup>T</sup> eγð Þ <sup>t</sup>�<sup>T</sup>

product cost is independent to xið Þt , i.e. Cið Þ¼ xið Þt Ci, then optimal promotional

xið Þt Xi � �

� � � �

1 þ μ<sup>T</sup> eγð Þ <sup>t</sup>�<sup>T</sup>

The optimal control promotional policy shows that when market is almost saturated, then differentiated market promotional expenditure rate and mass market promotional expenditure rate, respectively, should be zero (i.e. there is no need

For optimal control policy, the optimal sales trajectory using optimal values of

xið Þt Xi � �

1 þ μ<sup>T</sup> e<sup>γ</sup>ð Þ <sup>t</sup>�<sup>T</sup>

irrespective of their nature (i.e. whether equality or inequality). The Hamiltonian H of each of the segments is strictly concave in uið Þt and u tð Þ. According to the Mangasarian sufficiency theorem [22, 23], there exist unique values of promotional

<sup>i</sup> ð Þ<sup>t</sup> and <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> for each segment, respectively. From Eqs. (9)

<sup>∂</sup>ui � <sup>∂</sup>Ci ∂xi ∂xi <sup>∂</sup>ui x\_ <sup>i</sup>

∂ui

<sup>i</sup> and φ�<sup>1</sup> are inverse functions of ϕ<sup>i</sup> and φ, respectively. If we assume

� <sup>∂</sup>Ci ∂xi ∂xi <sup>∂</sup>ui x\_ <sup>i</sup>

Xi � 1 � gi � �xið Þ<sup>t</sup> � �

Xi � 1 � gi

� �xið Þ<sup>t</sup>

<sup>i</sup> ð Þ<sup>t</sup> � � and mass market promotional

¼ 0 (9)

<sup>∂</sup><sup>u</sup> <sup>¼</sup> <sup>0</sup> (10)

<sup>∂</sup>xið Þ<sup>t</sup> , <sup>λ</sup>ið Þ¼ <sup>T</sup> <sup>0</sup> (11)

<sup>∂</sup>W tð Þ, <sup>μ</sup>ð Þ <sup>T</sup> <sup>≥</sup>0, (12)

#### 4.1.1 When differentiated market promotional effort and mass market promotional effort costs are linear functions

Let us assume that differentiated market promotional effort and mass market promotional effort costs take the following linear forms: ϕið Þ¼ uið Þt κ<sup>i</sup> uið Þt and φð Þ¼ u tð Þ κ u tð Þ and ai ≤uið Þt ≤ Ai, a≤ u tð Þ≤ A, where ai, Ai, a, and A are positive constants which are minimum and maximum acceptable promotional effort rates (ai, Ai, a, and A are determined by the promotional budget) and κ<sup>i</sup> is the per unit cost of promotional effort per unit time towards i th segment and κ is the per unit cost of promotional effort per unit time towards mass market. Now, Hamiltonian can be defined as

$$H = \begin{pmatrix} \sum\_{i=1}^{M} [(P\_i - C\_i(\mathbf{x}\_i(t)) + \lambda\_i(t)) \ \dot{\mathbf{x}}\_i(t) - \kappa\_i \ u\_i(t)] - \kappa \, u(t) \\\\ -\mu(t) \left(\sum\_{i=1}^{M} \kappa\_i \, u\_i(t) + \kappa \, u(t)\right) \end{pmatrix} \tag{22}$$

Since Hamiltonian is linear in uið Þt and u tð Þ, optimal differentiated market promotional effort and mass market promotional effort as obtained by the maximum principle are given by

$$u\_i^\*(t) = \begin{cases} \overline{a}\_i \text{ if } B\_i \le 0\\ \overline{A}\_i \text{ if } B\_i > 0 \end{cases} \tag{23}$$

$$u^\*(t) = \begin{cases} \overline{a} \text{ if } D \le 0\\ \overline{A} \text{ if } D > 0 \end{cases} \tag{24}$$

where Bi ¼ ð Þ Pi � Ci þ λið Þt pi þ qi xi Xi Xi � <sup>1</sup> � gi xið Þ<sup>t</sup> � <sup>κ</sup>ið Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> and <sup>D</sup> <sup>¼</sup> <sup>∑</sup><sup>M</sup> <sup>i</sup>¼<sup>1</sup> <sup>α</sup>ið Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup> pi <sup>þ</sup> qi xi Xi Xi � <sup>1</sup> � gi xið Þ<sup>t</sup> � <sup>ε</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> are promotional effort switching functions and called 'bang-bang' control. However, interior control is possible on an arc along uið Þt and u tð Þ. Such an arc is known as the 'singular arc' [22].

u∗

u∗

x∗ <sup>i</sup> ðÞ¼ t

> x∗ <sup>i</sup> ðÞ¼ t

dλið Þt

<sup>φ</sup>ð Þ¼ u tð Þ <sup>κ</sup>

H ¼

u∗ <sup>i</sup> ðÞ¼ t

23

∑<sup>M</sup>

1 κi 0 @

0

B@

unit of variable xið Þt .

<sup>i</sup> ðÞ¼ <sup>t</sup> ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> A,ð3) <sup>u</sup><sup>∗</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81385

pi þqi xið Þ 0 Xi Xi� <sup>1</sup>�<sup>g</sup> ð Þ<sup>i</sup> xið Þ <sup>0</sup> !

� � pi

þqi xið Þ 0 Xi Xi� <sup>1</sup>�<sup>g</sup> ð Þ<sup>i</sup> xið Þ <sup>0</sup> !

If xið Þ¼ 0 0, then we get the following result

1 � exp � qi þ pi 1 � gi

dt <sup>¼</sup> ρλiðÞ�<sup>t</sup> ð Þ Pi � Ci <sup>þ</sup> <sup>λ</sup><sup>i</sup> Ai <sup>þ</sup> <sup>α</sup><sup>i</sup> <sup>A</sup> � � <sup>1</sup> � gi

pi

þ 1 � gi

1 � gi � � <sup>þ</sup> qi

the adjoint variable is given by

costs are quadratic functions

Xi

qi Xi <sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>a</sup>, and (4) <sup>u</sup><sup>∗</sup>

<sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>A</sup>, then the optimal sales and adjoint values can be described as

� � � � Ai <sup>þ</sup> <sup>α</sup><sup>i</sup> Ai

exp qi þ pi 1 � gi

� � � � Ai <sup>þ</sup> <sup>α</sup><sup>i</sup> Ai

which is similar to Bass model [7] sales trajectory with repeat purchasing, and

The value of λið Þt stands for per unit change in future profit of having one more

4.1.2 When differentiated market promotional effort and mass market promotional effort

Promotional efforts towards differentiated market and mass market are costly. Let us assume that differentiated market promotional effort and mass market pro-

<sup>2</sup> <sup>u</sup><sup>2</sup>ð Þ<sup>t</sup> where <sup>κ</sup><sup>i</sup> > 0 and <sup>κ</sup> > 0 are positive constants and represent the

<sup>2</sup> <sup>u</sup><sup>2</sup> <sup>i</sup>ð Þt

� �

� �xið Þ<sup>t</sup> � �

2 u2 ð Þt

mass market, respectively. This assumption is common in literature [24], where

h i � <sup>κ</sup>

�μð Þ<sup>t</sup> <sup>∑</sup><sup>M</sup>

From the optimality necessary conditions (6), the optimal differentiated market

xið Þt Xi � � Xi � <sup>1</sup> � gi

1 þ μ<sup>T</sup> e<sup>γ</sup>ð Þ <sup>t</sup>�<sup>T</sup>

i¼1 κi <sup>2</sup> <sup>u</sup><sup>2</sup> <sup>i</sup>ðÞþt κ 2 u2 ð Þt

motional effort costs take the following quadratic forms <sup>ϕ</sup>ið Þ¼ uið Þ<sup>t</sup> <sup>κ</sup><sup>i</sup>

magnitude of promotional effort rate per unit time towards i

promotion cost is quadratic. Now, Hamiltonian can be defined as

<sup>i</sup>¼<sup>1</sup> ð Þ Pi � Cið Þþ xið Þ<sup>t</sup> <sup>λ</sup>ið Þ<sup>t</sup> x t \_ðÞ� <sup>κ</sup><sup>i</sup>

promotional effort and mass market promotional effort are given by

ð Þ Pi � Ci þ λið Þt pi þ qi

� � � � Ai <sup>þ</sup> <sup>α</sup><sup>i</sup> Ai � �t � �

� �<sup>t</sup> � � , i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M (26)

� � 2x<sup>∗</sup>

� � � � , <sup>λ</sup>ið Þ¼ <sup>T</sup> <sup>0</sup>

<sup>i</sup> � Xi � � � Xi

and mass market promotional effort (u<sup>∗</sup>ð Þ<sup>t</sup> ) for each segment, we can obtain the optimal sales trajectories and adjoint trajectories. If we consider optimal values

Using these optimal values of differentiated market promotional effort (u<sup>∗</sup>

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase…

exp qi þ pi 1 � gi

� �<sup>t</sup> � � ! � pi

� � � � Ai <sup>þ</sup> <sup>α</sup><sup>i</sup> Ai � �t � �

exp � qi þ pi 1 � gi

<sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>A</sup> .

<sup>i</sup> ð Þt )

∀i ¼ 1, 2, 3, …, M

(25)

(27)

<sup>2</sup> u<sup>2</sup>

1

1

th segment and towards

<sup>i</sup>ð Þt and

CA (28)

A (29)

This optimal control advertising policy shows that when market is almost saturated, then our switching

$$\begin{array}{l} \text{functions } B\_{i} = (P\_{i} - C\_{i} + \lambda\_{i}(t)) \left( p\_{i} + q\_{i} \frac{\underline{\mathbf{x}}\_{i}}{\underline{\mathbf{x}}\_{i}} \right) \left( \overline{\mathbf{X}}\_{i} - \left( \mathbf{1} - \underline{\mathbf{g}}\_{i} \right) \mathbf{x}\_{i}(t) \right) - \kappa\_{i}(1 + \mu(t)) \text{ and} \\ D = \sum\_{i=1}^{M} \left( a\_{i} (P\_{i} - \mathbf{C}\_{i} + \lambda\_{i}(t)) \left( p\_{i} + q\_{i} \frac{\underline{\mathbf{x}}}{\underline{\mathbf{x}}\_{i}} \right) \left( \overline{\mathbf{X}}\_{i} - \left( \mathbf{1} - \underline{\mathbf{g}}\_{i} \right) \mathbf{x}\_{i}(t) \right) \right) - \varepsilon (1 + \mu(t)) \text{ become} \\ \text{negative or zero. Therefore, optimal adversarial policy shows that there is no need to depend money, time, or resources on adversarial, i.e. we do the adversary with} \\ \text{\"the universal uncertainty\" } \mathbf{x} \text{ is} \\ \text{\"o. } \overline{\mathbf{x}} \text{ is} \end{array}$$

minimum effectiveness rate. There are four possible sets of optimal control values of differentiated market promotional effort (u<sup>∗</sup> <sup>i</sup> ð Þ<sup>t</sup> ) and mass market promotional effort (u<sup>∗</sup>ð Þ<sup>t</sup> ) rate (Figures 1 and 2): (1) u<sup>∗</sup> <sup>i</sup> ðÞ¼ <sup>t</sup> ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> a, <sup>ð</sup>2)

Figure 1.

Optimal promotional effort allocation policy for mass market promotional effort.

Figure 2. Optimal promotional effort allocation policy for differentiated market promotional.

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase… DOI: http://dx.doi.org/10.5772/intechopen.81385

u∗ <sup>i</sup> ðÞ¼ <sup>t</sup> ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> A,ð3) <sup>u</sup><sup>∗</sup> <sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>a</sup>, and (4) <sup>u</sup><sup>∗</sup> <sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>A</sup> . Using these optimal values of differentiated market promotional effort (u<sup>∗</sup> <sup>i</sup> ð Þt ) and mass market promotional effort (u<sup>∗</sup>ð Þ<sup>t</sup> ) for each segment, we can obtain the optimal sales trajectories and adjoint trajectories. If we consider optimal values u∗ <sup>i</sup> ðÞ¼ <sup>t</sup> Ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> <sup>A</sup>, then the optimal sales and adjoint values can be described as

$$\begin{split} \mathbf{x}\_{i}^{\*}(\mathbf{t}) &= \frac{\overline{\mathbf{X}\_{i}}\left(\left(\frac{p\_{i} + q\_{i}\frac{\mathbf{x}\_{i}^{\*}}{\mathbf{X}\_{i}}}{\overline{\mathbf{X}\_{i}} - (\mathbf{1} - \mathbf{g}\_{i})\mathbf{x}\_{i}(\mathbf{0})}\right) \exp\left(\left(q\_{i} + p\_{i}(\mathbf{1} - \mathbf{g}\_{i})\right)\left(\overline{\mathbf{A}\_{i}} + a\_{i}\overline{\mathbf{A}\_{i}}\right)t\right)\right) - p\_{i}}{\frac{q\_{i}}{\mathbf{X}\_{i}} + \left(\mathbf{1} - \mathbf{g}\_{i}\right)\left(\frac{p\_{i} + q\_{i}\frac{\mathbf{x}\_{i}^{\*}}{\mathbf{X}\_{i}}}{\overline{\mathbf{X}\_{i}} - (\mathbf{1} - \mathbf{g}\_{i})\mathbf{x}\_{i}(\mathbf{0})}\right) \exp\left(\left(q\_{i} + p\_{i}(\mathbf{1} - \mathbf{g}\_{i})\right)\left(\overline{\mathbf{A}\_{i}} + a\_{i}\overline{\mathbf{A}\_{i}}\right)t\right)}\tag{25} \tag{25} \end{split} \tag{26}$$

If xið Þ¼ 0 0, then we get the following result

$$\mathbf{x}\_{i}^{\*}(t) = \frac{\mathbf{1} - \exp\left(-\left(q\_{i} + p\_{i}(\mathbf{1} - \mathbf{g}\_{i})\right)\left(\overline{A}\_{i} + a\_{i}\,\overline{A}\_{i}\right)t\right)}{\left(\mathbf{1} - \mathbf{g}\_{i}\right) + \frac{q\_{i}}{p\_{i}}\,\exp\left(-\left(q\_{i} + p\_{i}(\mathbf{1} - \mathbf{g}\_{i})\right)\left(\overline{A}\_{i} + a\_{i}\,\overline{A}\_{i}\right)t\right)}, i = \mathbf{1}, 2, ..., M \quad \text{(26)}$$

which is similar to Bass model [7] sales trajectory with repeat purchasing, and the adjoint variable is given by

$$\frac{d\lambda\_i(t)}{dt} = \rho \lambda\_i(t) - \left( (P\_i - \mathbf{C}\_i + \lambda\_i) \left( \overline{A}\_i + a\_i \,\overline{A} \right) \left( (\mathbf{1} - \mathbf{g}\_i) \left( 2\mathbf{x}\_i^\* - \mathbf{X}\_i \right) - \mathbf{X}\_i \right) \right), \lambda\_i(T) = \mathbf{0} \tag{27}$$

The value of λið Þt stands for per unit change in future profit of having one more unit of variable xið Þt .

#### 4.1.2 When differentiated market promotional effort and mass market promotional effort costs are quadratic functions

Promotional efforts towards differentiated market and mass market are costly. Let us assume that differentiated market promotional effort and mass market promotional effort costs take the following quadratic forms <sup>ϕ</sup>ið Þ¼ uið Þ<sup>t</sup> <sup>κ</sup><sup>i</sup> <sup>2</sup> u<sup>2</sup> <sup>i</sup>ð Þt and <sup>φ</sup>ð Þ¼ u tð Þ <sup>κ</sup> <sup>2</sup> <sup>u</sup><sup>2</sup>ð Þ<sup>t</sup> where <sup>κ</sup><sup>i</sup> > 0 and <sup>κ</sup> > 0 are positive constants and represent the magnitude of promotional effort rate per unit time towards i th segment and towards mass market, respectively. This assumption is common in literature [24], where promotion cost is quadratic. Now, Hamiltonian can be defined as

$$H = \begin{pmatrix} \sum\_{i=1}^{M} \left[ \left( P\_i - \mathbf{C}\_i(\mathbf{x}\_i(t)) + \lambda\_i(t) \right) \dot{\mathbf{x}}(t) - \frac{\kappa\_i}{2} \boldsymbol{u}\_i^2(t) \right] - \frac{\kappa}{2} \boldsymbol{u}^2(t) \\\\ -\mu(t) \left( \sum\_{i=1}^{M} \frac{\kappa\_i}{2} \left. \boldsymbol{u}\_i^2(t) + \frac{\kappa}{2} \left. \boldsymbol{u}^2(t) \right) \right) \end{pmatrix} \tag{28}$$

From the optimality necessary conditions (6), the optimal differentiated market promotional effort and mass market promotional effort are given by

$$u\_i^\*(t) = \frac{1}{\kappa\_i} \left( \frac{(P\_i - C\_i + \lambda\_i(t)) \left(p\_i + q\_i \frac{\mathbf{x}\_i(t)}{\mathbf{X}\_i}\right) (\overline{X\_i} - (\mathbf{1} - \mathbf{g}\_i)\mathbf{x}\_i(t))}{\mathbf{1} + \mu\_T \,\sigma^{r(t-T)}} \right) \tag{29}$$

where Bi ¼ ð Þ Pi � Ci þ λið Þt pi þ qi

<sup>i</sup>¼<sup>1</sup> <sup>α</sup>ið Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup> pi <sup>þ</sup> qi

functions Bi ¼ ð Þ Pi � Ci þ λið Þt pi þ qi

<sup>i</sup>¼<sup>1</sup> <sup>α</sup>ið Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup> pi <sup>þ</sup> qi

<sup>D</sup> <sup>¼</sup> <sup>∑</sup><sup>M</sup>

<sup>D</sup> <sup>¼</sup> <sup>∑</sup><sup>M</sup>

Figure 1.

Figure 2.

22

'singular arc' [22].

Industrial Engineering

rated, then our switching

minimum effectiveness rate.

promotional effort (u<sup>∗</sup>

(Figures 1 and 2): (1) u<sup>∗</sup>

xi Xi 

promotional effort switching functions and called 'bang-bang' control. However, interior control is possible on an arc along uið Þt and u tð Þ. Such an arc is known as the

> xi Xi

negative or zero. Therefore, optimal advertising policy shows that there is no need to spend money, time, or resources on advertising, i.e. we do the advertising with

There are four possible sets of optimal control values of differentiated market

xi Xi 

<sup>i</sup> ðÞ¼ <sup>t</sup> ai, u<sup>∗</sup>ðÞ¼ <sup>t</sup> a, <sup>ð</sup>2)

Optimal promotional effort allocation policy for mass market promotional effort.

Optimal promotional effort allocation policy for differentiated market promotional.

xi Xi  Xi � 1 � gi

Xi � 1 � gi

xið Þ<sup>t</sup>

Xi � 1 � gi

<sup>i</sup> ð Þ<sup>t</sup> ) and mass market promotional effort (u<sup>∗</sup>ð Þ<sup>t</sup> ) rate

� <sup>ε</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> become

xið Þ<sup>t</sup>

Xi � 1 � gi

� <sup>ε</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> are

This optimal control advertising policy shows that when market is almost satu-

xið Þ<sup>t</sup> � <sup>κ</sup>ið Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> and

xið Þ<sup>t</sup> � <sup>κ</sup>ið Þ <sup>1</sup> <sup>þ</sup> <sup>μ</sup>ð Þ<sup>t</sup> and

$$u(t) = \frac{1}{\kappa} \left( \frac{\sum\_{i=1}^{M} \left( (P\_i - \mathbf{C}\_i + \lambda\_i(t)) \right) a\_i \left( p\_i + q\_i \frac{\mathbf{x}\_i(t)}{X\_i} \right) \left( \overline{X\_i} - (\mathbf{1} - \mathbf{g}\_i) \mathbf{x}\_i(t) \right)}{\mathbf{1} + \mu\_T \, e^{r(t-T)}} \right) \tag{30}$$

Using optimal differentiated market promotional effort and mass market promotional effort rates from above Eqs. (29) and (30), we can obtain the optimal sales trajectories. Due to cumbersome analytical expression and an aim to illustrate the applicability of the formulated model through a numerical example, the discounted continuous optimal problem (5) is transformed into equivalent discrete problem [25] which can be solved using differential evolution. The equivalent discrete optimal control of the budgetary problem can be written as follows:

$$\mathbf{Max}\,J=\sum\_{k=1}^{T}\Biggl( \begin{pmatrix} \left[\sum\_{i=1}^{M}(P\_{i}-\mathbf{C}\_{i}(k))(\mathbf{x}\_{i}(k+1)-\mathbf{x}\_{i}(k)-\phi\_{i}(\mathbf{u}\_{i}(k))\right] \\\\ -\boldsymbol{\varrho}(\boldsymbol{u}(k)) \end{pmatrix} \bigg) \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \left(1+\boldsymbol{\gamma}\right)^{k-1} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \boldsymbol{\gamma} \end{pmatrix} \Bigg) \Bigg( \begin{pmatrix} 1 \\ \boldsymbol{\gamma} \end{pmatrix} \Bigg) \Bigg$$

$$(31)$$

The discrete optimal control problem developed in this chapter is solved using differential evolution. Total promotional budget is assumed to be ₹ 3,000,000,000 which has to be allocated for mass market promotion and segment-specific promotion in four segments of the market. The time horizon has been divided into 12 equal time periods. The number of market segments is four (i.e. M = 4). The

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase…

Optimal allocation of promotional effort resources by solving each segment is

S1 S2 S3 S4 T1 13.61 2.14 5.16 1.00 12.62 T2 14.69 1.30 2.42 1.00 13.20 T3 14.71 1.57 5.61 1.09 14.13 T4 17.06 6.87 1.56 1.50 15.02 T5 7.56 2.07 6.03 1.00 15.91 T6 19.63 2.32 1.96 1.00 16.80 T7 10.02 2.56 2.16 1.00 17.67 T8 21.99 4.88 6.62 3.06 16.61 T9 11.17 3.01 2.54 1.32 19.35 T10 24.28 8.27 7.00 2.74 20.19 T11 13.54 3.43 2.90 3.11 21.01 T12 26.44 8.68 7.34 10.74 28.78

S1 S2 S3 S4

T1 8969 8000 8000 8000 T2 33,386 25,338 39,111 20,804 T3 112,850 64,724 105,406 52,397 T4 292,429 142,818 112,330 124,026 T5 296,492 201,723 98,941 235,744 T6 291,852 59,131 135,301 217,610 T7 297,762 157,208 35,605 235,055 T8 289,245 155,499 106,680 216,804 T9 305,967 158,657 108,227 241,642 T10 276,207 151,979 103,967 203,831 T11 345,626 178,184 121,081 263,316 T12 178,359 90,396 61,631 131,523

Differentiated Mass

given in Table 2 for both mass and differentiated promotions, and the

problem is coded in Lingo11 and solved.

DOI: http://dx.doi.org/10.5772/intechopen.81385

corresponding sales is tabulated in Table 3.

Optimal differentiated and mass promotional allocations (in units).

Table 2.

Table 3.

25

Optimal sales from potential market.

The discretized version of the model is NP-hard; therefore, we use Lingo11 [26] to solve the discrete formulation.

#### 5. Numerical illustration

To validate the model formulation, we consider a case of a company that has to find the optimal advertising policies for its consumer durable product. The company advertises at both national and regional levels of the market. To find the advertising policy for four segments, the values of the parameters, price, and cost of the product are given in Table 1.


Table 1. Parameters. Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase… DOI: http://dx.doi.org/10.5772/intechopen.81385

The discrete optimal control problem developed in this chapter is solved using differential evolution. Total promotional budget is assumed to be ₹ 3,000,000,000 which has to be allocated for mass market promotion and segment-specific promotion in four segments of the market. The time horizon has been divided into 12 equal time periods. The number of market segments is four (i.e. M = 4). The problem is coded in Lingo11 and solved.

Optimal allocation of promotional effort resources by solving each segment is given in Table 2 for both mass and differentiated promotions, and the corresponding sales is tabulated in Table 3.


#### Table 2.

u tðÞ¼ <sup>1</sup> κ

Industrial Engineering

Max <sup>J</sup> <sup>¼</sup> <sup>∑</sup><sup>T</sup>

k¼1

xið Þ¼ k þ 1 xið Þþ k pi þ qi

to solve the discrete formulation.

the product are given in Table 1.

ε(in ₹) 1,153,922

Table 1. Parameters.

24

5. Numerical illustration

B@

∑<sup>M</sup>

0 B@

∑<sup>M</sup>

0 @

<sup>i</sup>¼<sup>1</sup> ð Þ Pi � Ci <sup>þ</sup> <sup>λ</sup>ið Þ<sup>t</sup> <sup>α</sup><sup>i</sup> pi <sup>þ</sup> qi

optimal control of the budgetary problem can be written as follows:

xið Þk Xi

� �

xið Þt Xi � �

� � � �

1 þ μ<sup>T</sup> e<sup>γ</sup>ð Þ <sup>t</sup>�<sup>T</sup>

Using optimal differentiated market promotional effort and mass market promotional effort rates from above Eqs. (29) and (30), we can obtain the optimal sales trajectories. Due to cumbersome analytical expression and an aim to illustrate the applicability of the formulated model through a numerical example, the discounted continuous optimal problem (5) is transformed into equivalent discrete problem [25] which can be solved using differential evolution. The equivalent discrete

> <sup>i</sup>¼1ð Þ Pi � Cið Þ <sup>k</sup> ð Þ xið Þ� <sup>k</sup> <sup>þ</sup> <sup>1</sup> xið Þ� <sup>k</sup> <sup>ϕ</sup>ið Þ uið Þ<sup>k</sup> h i

! <sup>0</sup>

ð Þ uið Þþ k α<sup>i</sup> u kð Þ Xi � 1 � gi

∑<sup>T</sup> <sup>k</sup>¼<sup>1</sup> <sup>∑</sup><sup>M</sup>

The discretized version of the model is NP-hard; therefore, we use Lingo11 [26]

To validate the model formulation, we consider a case of a company that has to find the optimal advertising policies for its consumer durable product. The company advertises at both national and regional levels of the market. To find the advertising policy for four segments, the values of the parameters, price, and cost of

N<sup>i</sup> 279106.6 152460.1 97580.78 215868.5 pi 0.000766 0.001161 0.00138 0.000549 qi 0.137605 0.480576 0.540395 0.31362 α<sup>i</sup> 0.3 0.19 0.189 0.320568 gi 0.05 0.0265 0.0878 0.047644 κ<sup>i</sup> (in ₹) 243,961 388,753 336,791 517,530

P<sup>i</sup> 400,000 440,000 420,000 450,000 C<sup>i</sup> 340,000 370,000 340,000 390,000 Initial sales<sup>i</sup> 8969 8000 8000 8000

S1 S2 S3 S4

Xi � 1 � gi

�φð Þ u kð Þ

1 CA

� �xið Þ<sup>k</sup> � �, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, M

<sup>i</sup>¼1ð Þþ <sup>ϕ</sup>ið Þ uið Þ<sup>k</sup> <sup>φ</sup>ð Þ u kð Þ � �

� �xið Þ<sup>t</sup>

1

1 ð Þ 1 þ γ

k�1

subjected to

≤ W<sup>0</sup>

(31)

1 CA 9

>>>>>>>>>>>>>=

>>>>>>>>>>>>>;

A (30)

Optimal differentiated and mass promotional allocations (in units).


#### Table 3. Optimal sales from potential market.

In the above case, we have solved the discretized problem by taking differentiated and mass promotional efforts as a linear function.

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DOI: http://dx.doi.org/10.5772/intechopen.81385

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