Open access peer-reviewed chapter

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase in Segmented Market: A Control Theoretic Approach

By Kuldeep Chaudhary and Prakash C. Jha

Submitted: March 9th 2018Reviewed: September 7th 2018Published: November 14th 2018

DOI: 10.5772/intechopen.81385

Downloaded: 180

Abstract

This chapter considers an optimal control model to obtain dynamic promotional policies for a product considering a segmented market where first-time and additional repeat purchase sales are assumed to be generated through mass and differentiated promotions. Mass promotion is carried out in the whole market which reaches each segment with a fixed spectrum, and differentiated promotion is catered to each segment individually. The firm’s finite promotional resources are to be allocated for promoting a product at mass and segment levels of the market in a finite time period. The formulated control problem obtains optimal promotional effort policy for each segment using the maximum principle. The applicability of the proposed control model is illustrated through a numerical example by discretizing the model.

Keywords

  • market segmentation
  • innovation-diffusion model
  • optimal control theory
  • maximum principle

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.

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.

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, 2, 3]. A discrete time stochastic model of multiple media selection in a segmented market was analysed by Little and Lodish [1]. 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, 9, 10, 11, 12, 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 M>1is the total market segments and a discrete variable. The sum i=1MXi¯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 xitbe the number of adopter by time t for the ith segment. During diffusion process, 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:

dxitdt=bituit+αiutXi¯1gixit,i=1,2,,ME1

with the initial condition xi0=xi0i=1,2,,M,where αidenotes the segment spectrum of mass promotion αi>0&i=1Mαi=1; gi0gi1is susceptible to repeat purchasing, and repeat purchasing is influenced by all factors (both internal and external) affecting first purchase in ith segment by time t; uitis differentiated promotional effort rate for ith segment at time t; andutis mass market promotional effort rate at time t, and bitis the adoption rate per additional adoption for the ith segment. bitcan 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:

dxitdt=pi+qixitXi¯uit+αiutXi¯1gixit,i=1,2,,ME2

where pi and qi are coefficients of external and internal influences in ith 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

MaxJ=0Teγti=1M[PiCixitẋtϕiuitφutdtE3

where ϕiuitand φutare differentiated market promotional effort and mass market promotional effort cost, respectively, γis discounted profit, Piis sales price for ith segment, and Cixitis production cost per unit for ith segment, that is, continuous and differentiable with assumption Ci.>0and PiCixit>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

0Ti=1Mϕiuit+φutdtW0E4

where W0is 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:

MaxJ=0Teγti=1M[PiCixitẋtϕiuitφutdtdxitdt=pi+qixitXi¯uit+αiutXi¯1gixit,i=1,2,,Mxi0=xi0i=1,2,M0Ti=1Mϕiuit+φutdtW0E5

The above formulated optimal control problem consists of 2M+1control variables uitutand Mstate variables xit.

4. Solution approach

To solve the above optimal control theory problem, we define a new state variable Wt=W00ti=1Mϕiuit+φutdtwith W0=W0and WT0. With new state variable, we rewrite the above optimal control problem (5) as

MaxJ=0Teγti=1MPiCixitẋtϕiuitφutdtdxitdt=pi+qixitXi¯uit+αiutXi¯1gixit,i=1,2,,Mxi0=xi0i=1,2,MẆt=i=1Mϕiuit+φut,W0=W0,WT0E6

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

H=i=1MPiCixit+λitẋtϕiuitφutμti=1Mϕiuit+φutE7

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 λitand μtfor all t0T. The value of λitand μtdefine marginal valuation of state variables xitand Wtat time t, respectively. Here,λitstands for change in future profit as making a small in xitat time t, and μtis 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

Htxiuiuλμ=HtxiuiuλμE8
Hui=0E9
Hu=0E10
dλitdt=γλitHxit,λiT=0E11
tdt=λμtHWt,μT0,E12
WT+W00,μTWT+W0=0E13

Here, μT0,WT+W00,μTWT+W0=0are called as transversality conditions for Wt. Here, Hamiltonian is independent to Wt, and then we have μ̇=γμHWμt=μTeγtT. 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 uitand ut. According to the Mangasarian sufficiency theorem [22, 23], there exist unique values of promotional effort controls uitand utfor each segment, respectively. From Eqs. (9) and (10), we get

uit=ϕi1PiCixit+λitxi̇tuiCixixiuiẋi1+μTeγtT,i=1,2,,ME14
ut=φ1i=1MPiCixit+λitxi̇tuiCixixiuiẋi1+μTeγtT,i=1,2,,ME15

where ϕi1and φ1are inverse functions of ϕiand φ, respectively. If we assume product cost is independent to xit,i.e. Cixit=Ci, then optimal promotional effort policies for each segment become

uit=ϕi1PiCi+λitpi+qixitXi¯Xi¯1gixit1+μTeγtT,i=1,2,,ME16
ut=φ1i=1MPiCi+λitαipi+qixitXi¯Xi¯1gixit1+μTeγtT,i=1,2,,ME17

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 uitand mass market promotional effort utrates for each segment are given by

xit=Xi¯pi+qixi0Xi¯Xi¯1gixi0expqi+pi1gi0tuiτ+αiuτpiqiXi¯+1gipi+qixi0Xi¯Xi¯1gixi0expqi+pi1gi0tuiτ+αiuτE18

If xi0=0, then we get the following result:

xit=1expqi+pi1gi0tuiτ+αiuτ1gi+qipiexpqi+pi1gi0tuiτ+αiuτ,i=1,2,,ME19

and adjoint trajectory is given as

dλitdt=γλitPiCixit+λitxi̇xixi̇tCixitxitE20

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

λit=eγttTeγsPiCi+λitxi̇xiẋitCixidtE21

4.1 Particular cases

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: ϕiuit=κiuitand φut=κutand a¯iuitA¯i, a¯utA¯, where a¯i,A¯i,a¯, and A¯are positive constants which are minimum and maximum acceptable promotional effort rates (a¯i,A¯i,a¯,and A¯are determined by the promotional budget) and κiis the per unit cost of promotional effort per unit time towards ith segment and κis the per unit cost of promotional effort per unit time towards mass market. Now, Hamiltonian can be defined as

H=i=1MPiCixit+λitxi̇tκiuitκutμti=1Mκiuit+κutE22

Since Hamiltonian is linear in uitand ut, optimal differentiated market promotional effort and mass market promotional effort as obtained by the maximum principle are given by

uit=a¯iifBi0A¯iifBi>0E23
ut=a¯ifD0A¯ifD>0E24

where Bi=PiCi+λitpi+qixiXi¯X¯i1gixitκi1+μtand D=i=1MαiPiCi+λitpi+qixiXi¯X¯i1gixitε1+μtare promotional effort switching functions and called ‘bang-bang’ control. However, interior control is possible on an arc along uitand ut. Such an arc is known as the ‘singular arc’ [22].

This optimal control advertising policy shows that when market is almost saturated, then our switching functions Bi=PiCi+λitpi+qixiXi¯X¯i1gixitκi1+μtand D=i=1MαiPiCi+λitpi+qixiXi¯X¯i1gixitε1+μtbecome 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 minimum effectiveness rate.

There are four possible sets of optimal control values of differentiated market promotional effort (uit) and mass market promotional effort (ut) rate (Figures 1 and 2): (1) uit=a¯i,ut=a,¯(2) uit=a¯i,ut=A¯,(3)uit=A¯i,ut=a¯, and (4) uit=A¯i,ut=A¯. Using these optimal values of differentiated market promotional effort (uit) and mass market promotional effort (ut) for each segment, we can obtain the optimal sales trajectories and adjoint trajectories. If we consider optimal values uit=A¯i,ut=A¯, then the optimal sales and adjoint values can be described as

xit=Xi¯pi+qixi0Xi¯Xi¯1gixi0expqi+pi1giA¯i+αiA¯itpiqiXi¯+1gipi+qixi0Xi¯Xi¯1gixi0expqi+pi1giA¯i+αiA¯iti=1,2,3,,ME25

Figure 1.

Optimal promotional effort allocation policy for mass market promotional effort.

Figure 2.

Optimal promotional effort allocation policy for differentiated market promotional.

If xi0=0, then we get the following result

xit=1expqi+pi1giA¯i+αiA¯it1gi+qipiexpqi+pi1giA¯i+αiA¯it,i=1,2,,ME26

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

dλitdt=ρλitPiCi+λiA¯i+αiA¯1gi2xiXiXi,λiT=0E27

The value of λitstands for per unit change in future profit of having one more unit of variable xit.

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 ϕiuit=κi2ui2tand φut=κ2u2twhere κi>0and κ>0are positive constants and represent the magnitude of promotional effort rate per unit time towards ith 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=i=1MPiCixit+λitẋtκi2ui2tκ2u2tμti=1Mκi2ui2t+κ2u2tE28

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

uit=1κiPiCi+λitpi+qixitXi¯Xi¯1gixit1+μTeγtTE29
ut=1κi=1MPiCi+λitαipi+qixitXi¯Xi¯1gixit1+μTeγtTE30

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:

MaxJ=k=1Ti=1MPiCikxik+1xikϕiuikφuk11+γk1subjected toxik+1=xik+pi+qixikXi¯uik+αiukXi¯1gixik,i=1,2,,Mk=1Ti=1Mϕiuik+φukW0E31

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.

S1S2S3S4
N¯i279106.6152460.197580.78215868.5
pi0.0007660.0011610.001380.000549
qi0.1376050.4805760.5403950.31362
αi0.30.190.1890.320568
gi0.050.02650.08780.047644
κi (in ₹)243,961388,753336,791517,530
ε(in ₹)1,153,922
Pi400,000440,000420,000450,000
Ci340,000370,000340,000390,000
Initial salesi8969800080008000

Table 1.

Parameters.

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.

DifferentiatedMass
S1S2S3S4
T113.612.145.161.0012.62
T214.691.302.421.0013.20
T314.711.575.611.0914.13
T417.066.871.561.5015.02
T57.562.076.031.0015.91
T619.632.321.961.0016.80
T710.022.562.161.0017.67
T821.994.886.623.0616.61
T911.173.012.541.3219.35
T1024.288.277.002.7420.19
T1113.543.432.903.1121.01
T1226.448.687.3410.7428.78

Table 2.

Optimal differentiated and mass promotional allocations (in units).

S1S2S3S4
T18969800080008000
T233,38625,33839,11120,804
T3112,85064,724105,40652,397
T4292,429142,818112,330124,026
T5296,492201,72398,941235,744
T6291,85259,131135,301217,610
T7297,762157,20835,605235,055
T8289,245155,499106,680216,804
T9305,967158,657108,227241,642
T10276,207151,979103,967203,831
T11345,626178,184121,081263,316
T12178,35990,39661,631131,523

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.

6. Conclusion

This chapter formulates an optimal control problem to find the optimal promotional policies for a consumer durable product in a segmented market where the sales are evolved through the combination of two promotion strategies: mass and differentiated promotions. The sales include the first-time purchase and the repeat purchases built through loyalty towards the product. Also to make the problem more realistic, we take a total budget constraint. The objective is to maximize the total profit through promotion. Maximum principle has been used to obtain the solution of the proposed problem. After discretizing the problem with linear costs, a numerical example has been solved using Lingo11 to illustrate the applicability of the approach. The developed optimal control model can be further extended in several ways. For instance, factors such as price, quality, and cost can be incorporated along with differentiated and mass market promotional effort expenditures. Further this monopolistic model can also be extended to competitive duopolistic or oligopolistic markets. Also the model can be extended to obtain optimal control policies for two and/or more generations’ products in the market.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this chapter.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kuldeep Chaudhary and Prakash C. Jha (November 14th 2018). Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase in Segmented Market: A Control Theoretic Approach, Industrial Engineering, Ainul Akmar Mokhtar and Masdi Muhammad, IntechOpen, DOI: 10.5772/intechopen.81385. Available from:

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