A model for communication in a multi-segment market

A model for communication in a multi-segment market

Igor BYKADOROV*, Andrea ELLERO** and Elena MORETTI**

*Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences **Department of Management, Università Ca'Foscari Venezia

Abstract. We consider a linear optimal control model for the marketing of seasonal products which are produced by the same firm and sold by retailers in different market segments The horizon is divided in two consecutive non-intersecting intervals, called production and selling periods, respectively. The production period state variables are the inventory levels and two kinds of goodwills (consumers' and retailers' goodwill, respectively) while the selling period state variables are the sales levels and the two kinds of goodwills. In the production interval there are three kinds of controls: on production, quality and advertising, while in the selling one the controls are on communication via advertising, promotion ad-dressed to consumers and incentives given to retailers. We consider the case of several kinds of commu-nications. The optimal control problem is transformed into an equivalent nonlinear programming prob-lem.

Keywords. Optimal control, advertising, seasonal products, segmentation. .

1. Introduction

In this paper we consider a linear optimal control model for the marketing of seasonal products which are produced by the same firm and sold by retailers in differ-ent market segmdiffer-ents. More precisely, the firm produces and sells a seasonal product with different attributes in two consecutive time intervals, the first one devoted to production and the second one to the sale of the product. In the production period the firm can control production, product quality and advertise the product both to retailers and to the different market segments. In the selling period the firm has a wider communication activity via promo-tion to consumers, incentives to retailers and advertising to both. We assume that the communication efforts of the firm act on the goodwill that consumers and retailers as-sign to the product. The motion equations of the model refer explicitly to the consumers' and retailers' goodwills as state variables. The firm seeks the maximum profit but has also to keep a minimum level of goodwills at the end of the selling period. This way we generalize the linear models proposed in [1]-[3] (see also [4]-[6]).

2. A multi-segment linear model

We consider a model with

n

market segments,

r

retailers and

d

kinds of communications. As in [2], we assume that production and sales take place in two dis-tinct intervals

[

t

0

,

t

1

]

(production period) and

[

t

1

,

t

2

]

selling period). We will use the index sets

}.

,

1

{

},

,

1

{

},

,

,

1

{

n

J

r

K

d

I



_



_



_

2.1. State variables and controls

Through the production period we consider

n

state variables



)

(t

m

_i inventory level in the

i

-th segment,

i



I

, at time

t



[

t

₀

,

t

₁

]

,

and

n



1

controls



(t)

u

_i production expenditure rate in the

i

-th seg-ment,

i



I

, at time

t



[

t

₀

,

t

₁

]

,



q

expenditure rate for improving quality; as in the one-segment linear model [2],

q

will be consid-ered constant and nonnegative.

Through the selling period the state variables are



(t)

x

_i sales level in the

i

-th segment,

i



I

, at time

]

,

[

t

₁

t

₂

t



.

Moreover, through both production and selling periods we consider the state variables



(t)

C

_i consumer goodwill in the

i

-th segment,

i



I

, at time

t



[

t

₀

,

t

₂

]

,



(t)

R

_j goodwill of the j-th retailer,

j



J

, at time

]

,

[

t

₀

t

₂

t



, and

d

controls



)

(t

a

k expenditure rate in the

k

-th type of

commu-nication,

k



K

, at time

t



[

t

0

,

t

2

]

.

The consumer goodwills

C

_i

(t)

explain how easily the consumers in the

i

-th segment choose to buy the product, while the goodwills

R

j

(t)

take into account how

fa-vorably the

j

-th retailer keeps the product in stock (see [2]). For what concerns controls

a

_k

(t

)

, we consider that during the production period the communication co-4, Acad. Koptyug avenue, Novosibirsk, 630090, Russia,

Phone +7 383-363-46-06, Fax: +7 383-333-25-98, e-mail: bykad@math.nsc.ru

incides with advertising, while in the selling period ad-vertising, promotion and incentives are all available.

2.2. Dynamics

The dynamics of the model is rather different in the pro-duction period

[

t

₀

,

t

₁

]

and in the selling period

[

t

₁

,

t

₂

]

. The dynamics in the production period

[

t

₀

,

t

₁

]

is de-scribed by

2n



r

motion equations:

n

equations for the inventory levels

m

i

(t

)

,

n

equations describing the

goodwills

C

_i

(t)

of the consumers' segments and

r

equations for the goodwills

R

j

(t)

of the retailers.

For the

i

-th segment,

i



I

, the inventory level dynamics is the following

),

(

)

(

t

u

t

m

_

_i





_{i i} where



i



marginal productivity of production expenditure,

1



i



.

Further, in the

i

-th segment,

i



I

, the goodwill level dynamics is given by the motion equations











K k k p k C i C i

t

C

t

a

t

C

i i

(

)

(

),

)

(





( )



where



i C



decay rate of goodwill

C

_i,



0

i C



,



) ( p k Ci



marginal productivity of expenditure in

commu-nication

a

_k, via advertising, in terms of goodwill

i

C

in the production period, _C(p_k)



0

i



.

Remark that _C(p_k)



0

i



means that, in the production

pe-riod, communication expenditure

a

k for advertising

in-creases the goodwill

C

i.

Finally, for the

j

-th retailer,

j



J

, the goodwill level dynamics is given by the motion equations

),

(

)

(

)

(

t

R

t

( )

a

t

R

_k K k p k R j R j j



j 













where



j R



decay rate of goodwill

R

_j

,



0

j R



,



) ( p k R_j



marginal productivity of expenditure in

commu-nication

a

_kvia advertising, in terms of goodwill

R

_j in the production period, _R(p_k)



0

j



.

The dynamics in the selling period

[

t

₁

,

t

₂

]

is also de-scribed by

2n



r

motion equations:

n

equations for sales levels

x

_i

(t)

in

i

-th segment and, as in the produc-tion period,

n

equations for goodwill

C

i in

i

-th

seg-ment and

r

equations for goodwill

R

_jof

j

-th retailer.

For the

i

-th segment,

i



I

, the inventory level dynam-ics is the following















J j j R x i C i i i

t

x

t

C

t

R

t

x

j i i

(

)

(

)

)

(

)

(









,

)

(

1 ) (

q

l

t

a

i i x K k k s k x











where



i



saturation aversion parameter,



_i



0

,



i

C



goodwill productivity in terms of sales originated by goodwill

C

_i,



0

i C



,



j iR x



goodwill productivity in terms of sales in

i

-th segment originated by the

j

-th retailer goodwill

j

R

,



0

j iR x



,



) (s k xi



marginal productivity of expenditure in

commu-nication

a

_k via promotion in terms of sales level

x

_i during the selling period, (_xs_k)



0

i



,



i

x

l

marginal sale productivity of the expenditure rate

q

for improving quality,



0

i x

l

,











_{1 0} 1

q

t

t

q

the total expenditure for improving

quality in the production period.

Further, for

i

-th segment,

i



I

, the goodwill level dynamics is given by the motion equation













K k C k s k C i C i i i

t

x

t

C

t

a

t

l

q

C

i i i

(

)

(

)

,

)

(

)

(







( ) ₁



where



i



word-of-mouth in terms of goodwill

C

i,



i



0

,



i

C



decay rate of goodwill

C

i,



C_i



0

,



) (s

k Ci



marginal productivity of expenditure in

commu-nication

a

k via advertising in terms of goodwill i

C

in the sales period, _C(s)_k



0

i



,



i

C

l

marginal productivity of quality expenditure rate

q

for goodwill

C

_i,



0

i

C

l

.

Finally, for the

j

-th retailer,

j



J

, the goodwill level dynamics is given by the motion equation













K k R k s k R j R j

t

R

t

a

t

l

q

R

j j j

(

)

(

)

,

)

(





( ) ₁



where



j R



decay rate of goodwill

R

j,



R_j



0

,



) ( s

k Rj



marginal productivity of expenditure in

commu-nication

a ,

_k via advertising and incentives, in terms of goodwill

R

j in the sales period,

0

) (s



k R_j



j

R

l

marginal productivity of

q

for goodwill

R

_j

,

0



j R

l

.

2.3. Objective function

The total profit of the firm is determined as the difference between the total income and the total expenditures (for production, product quality and communication) and is given by the following objective function:















_

_











  

dt

q

t

u

t

m

c

t

x

p

t t i I i I i i i I i i i 1 0

)

(

)

(

)

(

2

 

_



2 0

,

)

(

t t k K k

t

dt

a

where



i

p

product sale price in

i

-th segment,

p

_i



0

,



i

c

inventory marginal cost per time unit in

i

-th seg-ment,

c

_i



0

.

2.4. Boundary conditions

The natural boundary conditions for the inventory and sales levels are

.

),

(

)

(

,

0

)

(

)

(

t

0

x

t

1

x

t

2

m

t

1

i

I

m

i



i



i



i



The boundary conditions for the consumer goodwills are

,

,

)

(

,

)

(

t

₀

C

0

C

t

₂

C

2

i

I

C

_i



_{i i}



_i



where



0 i

C

initial value of goodwill

C ,

i



2 i

C

lower bound of goodwill

C

i at final time

t

2.

The boundary conditions for the retailers' goodwills are

,

,

)

(

,

)

(

t

₀

R

0

R

t

₂

R

2

j

J

R

_j



_{j j}



_j



where



0 j

R

initial value of goodwill

R

_j

,



2 j

R

lower bound of goodwill

R

jat final time

t

2.

The boundary conditions for the control variables are

],

,

[

,

],

,

0

[

)

(

t

u

i

I

t

t

0

t

1

u

i



i





],

,

[

,

],

,

0

[

)

(

t

a

k

K

t

t

0

t

2

a

k



k





].

,

0

[ q

q



where



i

u

upper bound for

u

_i

(t

)

,

u

_i



0

,



k

a

upper bound for

a

_k

(t

)

,

a

_k



0

,



q

upper bound for

q

,

q



0

.

2.5. Formulation of the model

In order to formulate the linear model we consider

n

-dimensional state variables vectors

m

(

t

),

x

(

t

),

C

(

t

)

with elements

m

_i

(

t

),

x

_i

(

t

),

C

_i

(

t

),

i



I

, respectively;

r

-dimensional state variables vector

R

(t

)

with elements

R

j

(

t

),

j



J

;

n

-dimensional control vector

)

(t

u

with elements

u

i

(

t

),

i



I

;

r

-dimensional control vector

a

(t

)

with elements

K

k

t

a

_k

(

),



;

n

-dimensional constant vectors

p

,

c

,

C

0

,

l

x

,

l

C

,

u

C ,

2 with elements

p

_i

c

_i

C

_i

l

_x

l

_C

C

_i

u

_i

i

I

i i

,

,

,

,



,

,

,

0 2 , respectively;

r

-dimensional constant vectors

R

0

,

l

R

,

R

2 with elements

R

j

,

l

R_j

,

R

j

,

j



J

2 0

, respectively;

d

-dimensional constant vector

a

with elements

K

k

a

_k

,



;

diagonal constant matrices



,



_C

,



,



_C

,



of order

n

with diagonal elements



i

,



C_i

,



,



C_i

,



i

,

I

i



, respectively;

diagonal constant matrix



_R of order

r

with

diagonal elements

j

J

j

R

,





;

constant

n



d

matrices



_C(p)

,



_x(s)

,



_C(s) with elements _Cp_{k x}s_{k C}s_k

i

I

k

K

i i i

,

,

,

 ,



) ( ) ( ) (







, respectively;

constant

r



d

matrices



_R(p)

,



_R(s) with elements

K

k

J

j

s k R p k Rj

,

j

,

 ,



) ( ) (





, respectively;

constant

n



r

matrix



_R with elements

J

j

I

i

j iR x

,

 ,





.

Moreover, let us define

],

,

[

,

)

(

)

(

)

(

],

,

[

,

)

(

)

(

)

(

_{0 2}

t

t

₁

t

₂

t

A

t

x

t

X

t

t

t

t

R

t

C

t

A

_































,

,

2 2 2 0 0 0





























R

C

A

R

C

A

, , 0 0 ) ( ) ( ) (                _p R p C p R rn nr C E









,

,

,

0

0

0

) ( ) ( ) ( ) (









































































R C x s R s C s x s R rn rn nr C R C

l

l

l

L

E

Q



















where, as usual,

0

nr and

0

rn are



n



r



and,

respec-tively,



r



n



zero matrices.

Moreover, let

e

n and

e

d be

n

-dimensional and,

respectively,

d

-dimensional unit vectors while

0

n and d

This way the linear marketing model with

n

seg-ments,

r

retailers and

d

kinds of communications is

Problem

P

: maximize

















1 0 2 0

,

)

(

)

(

)

(

)

(

2 t t t t T d T n T T

dt

t

a

e

dt

q

t

u

e

t

m

c

t

x

p

subject to

],

[

),

(

)

(

)

(

),

(

)

(

t

u

t

A

t

A

t

E

( )

a

t

t

t

_0,

t

₁

m

















p



,

)

(

,

0

)

(

t

₀

A

t

₀

A

0

m



_n



],

,

[

,

)

(

)

(

)

(

1 1 2 ) (

t

t

t

L

q

t

a

E

t

QX

t

X









s





,

)

(

),

(

)

(

,

0

)

(

2 2 1 2 1

x

t

m

t

A

t

A

t

x



_n





].

,

0

[

,

)

(

0

,

)

(

0

_n



u

t



u

_d



a

t



a

q



q

Remark that it is possible to replace the vector inequality

)

(

)

(

t

₂

m

t

₁

x



with an equality. In fact, if by

contradic-tion in an optimal solucontradic-tion of

P

x

(

t

₂

)



m

(

t

₁

)

, it would be possible to decrease production and, at the same time, increase the value of the objective function, due to the special form of the motion equations.

The model proposed is rather general and allows different specifications. One of them is the following. Assume the aim of the firm is to determine the optimal expenditure rate for each segment and each retailer. Then

r

n

d





. Moreover, let each kind of communication

act only on a single retailer or on a single segment of the market. It means that for every

i



I

and

k



K

\ i

{

}

one has _C(p_k)



_x(s_k)



_C(s_k)



0

i i i







and that for every

J

j



and

k



K

\

{

n



j

}

one has _R(p_k)



_R(s)_k



0

j j





.

Another possible interpretation of the model will be given in Conclusions.

3. Decomposition into parametric

subproblems

To discuss an approach for solving problem

P

, let us define

n

-dimensional vector

m

~

, whose elements

m

~

_i

,

I

i



, are parameters;

(

n



r

)

-dimensional vectors

A,

~

A

, whose elements

J

j

A

A

I

i

A

A

i i



nj

,

nj

,



~

,

,

,

~

, respectively, are pa-rameters too .

As in [3] problem

P

can be solved by first solving the optimal control problems, depending on the parame-ters

m

~



m

(

t

₁

),

A

~



A

(

t

₁

),

A



A

2 and then solving a nonlinear programming problem in which the parameters are the decision variables. Denote

.

~

,

~

0

~





























A

m

X

A

X

n

The parametric subproblems

P

₁



P

(

m

~

₁

)

,

P

2



P

2

(

A

~

)

and

P

₃



P

₃

(

m

,

q

,

A

~

,

A

)

can be written this way Problem

P

₁: maximize











1 0

,

)

(

)

(

t t T n T

dt

t

u

e

t

m

c

subject to

),

(

)

(

t

u

t

m







.

)

(

0

,

~

)

(

,

0

)

(

t

₀

m

t

₁

m

u

t

u

m



_n



_n





Problem

P

₂: maximize





1 0

,

)

(

t t T d

a

t

dt

e

subject to

),

(

)

(

)

(

t

A

t

E

( )

a

t

A











p

.

)

(

0

,

~

)

(

,

)

(

t

₀

A

0

A

t

₁

A

a

t

a

A





_d





Problem

P

3: maximize





2 1

,

)

(

t t T d

a

t

dt

e

subject to

,

)

(

)

(

)

(

1 ) (

L

q

t

a

E

t

QX

t

X









s



.

)

(

0

,

)

(

,

~

)

(

t

₁

X

X

t

₂

X

a

t

a

X





_d





Moreover, problem

P

₁ , due to the special form of its motion equations, can be replaced by

n

simpler prob-lems

P

i



P

(i)

(

m

~

_i

),

i



I

1 ) ( 1 , this way Problem 1() i

P

: maximize











1 0

,

)

(

)

(

t t i i i

m

t

u

t

dt

c

subject to

),

(

)

(

t

u

t

m



i





i i

].

,

0

[

)

(

,

~

)

(

,

0

)

(

0 i 1 i i i i

t

m

t

m

u

t

u

m







Let

F

~

₁(i)



F

~

₁(i)

(



~

_i

),

i



I

,

F

~

₂



F

~

₂

(

A

~

)

and

F

~

₃



)

,

~

,

,

(

~

3

m

q

A

A

F



be the optimal values of problems

,

,

) (

1

i

I

P

i



P

₂and

P

₃ respectively. Then problem

P

is equivalent to a nonlinear programming problem where the following objective function

p

m

~



t

₁

t

₀



q

F

~

₁()

F

~

₂

F

~

₃ I i i T













 (1) must be maximized. In the following we will assume that the general position condition (GPC) ([7], p.166) holds.

The optimal solutions of problems

P

1()

,

i

I

,

i



coincide with the ones for the case

n



1

(see [3]), i.e. for every

i



I

the optimal solution of problem

P

₁(i) is

,

)

,

(

,

)

,

(

,

0

)

(

1 0 *















t

t

t

u

t

t

t

t

u

i i u i u i

,

)

,

(

),

(

)

,

(

,

0

)

(

1 0 *

















t

t

t

t

t

u

t

t

t

t

m

i i i u u i i u i



where i i i u

u

m

t

t

i



~

1





. The optimum value is

 

,

~

2

)

~

(

~

() 2 1 i i i i i i i i

m

m

u

c

m

F









(2) where

m

~

_i



[

0

,



_i

u

_i

(

t

₁



t

₀

)].

(3) In problem

P

₂, due to GPC, the number of switches in the optimal control

a

*k

(

t

)



k



K

cannot

be more than

n



r

([7], p.166). Denote these switching times by



₁(k)

,

_



_n(k_)_r



k



K

. Let

t

₀





₁(k)



_





_n(k_)_r





_n(k_)_r1

,

k



K

,

(4) and denote

,

)

(

,

~

0 ( ) 1 ( ) ) ( 1 0 0 p k t p k t t p

E

e

a

t

D

A

e

A

e

G

















where

E

_k( p) is

k

-th column of matrix

E

(p). One has

























 

,

),

1

(

,

)

(

) ( ) ( ) (

odd

is

r

n

r

n

d

even

is

r

n

r

n

d

G

K k p k K k p k p (5) where

}.

1

,

{

,

)

(

)

1

(

)

(

1 ) ( ) ( ) (

















r

n

r

n

l

D

l

d

l j k j p k j p k



Moreover,

(

n



r

)

-dimensional vector

v

exists such that

( )

1

,

,

{

1

,

},

) (

r

n

j

K

k

E

e

v

T _kp k j 











 (6) and the optimum value of the objective function is

























 

,

),

1

(

,

),

(

~

) ( ) ( 2

odd

is

r

n

r

n

T

even

is

r

n

r

n

T

F

K k p k K k p k (7) where

}.

1

,

{

,

)

1

(

)

(

1 ) ( ) (



















r

n

r

n

l

a

l

T

l j k j j k p k



Now consider problem

P

3. For sake of simplicity we will

consider only the case where all eigenvalues of matrix

Q

are distinct. If some of them coincide, similar calculations can be made, as in [2]. Let

S

be the matrix of eigenvec-tors of matrix

Q

. Let us denote

.

}

,

,

,

{

1 2 2

S

1

QS

diag

n r  











_



Due to GPC, since the eigenvalues of the motion

equa-tions matrix are real, then



k



K

the number of

switches in the optimal control

a

*_k

(

t

)

cannot be more than

2

n



r

. Let us denote the switching times by

) ( 2 ) ( 1

,

,

k r n k 





_

. Let

,

,

2 ) ( 1 2 ) ( 2 ) ( 1 1

t

k

K

t

kn r k r n k















_



_



_{ } (8) and denote

,

)

(

~

1 1 1 1 ) (

_e

2

_S

_X

_e

1

_S

_X

_e

1

_e

2

_S

_L

G

s



t 



t 



t



t



 

,

,

)

(

1 1 ( ) ) (

K

k

E

S

e

a

t

D

_ks



_k t



  _ks



where

E

_k( s) is

k

-th column of matrix

E

(s).

Let

M



I

be the number of negative eigenval-ues of matrix

Q

.

Let

M

be even. Then





















 

,

,

)

1

2

(

,

),

2

(

) ( ) ( ) (

odd

is

r

r

n

d

even

is

r

r

n

d

G

K k s k K k s k s (9) where

}.

1

2

,

2

{

,

)

(

)

1

(

)

(

1 ) ( ) ( ) (

















r

n

r

n

l

D

l

d

l j k j s k j s k



Further, there exists a

(

2

n



r

)

-dimensional vector

w

such that

},

2

,

1

{

,

,

1

) ( 1 ) (

r

n

j

K

k

E

S

e

w

T _ks k j  











 (10) and the optimum value of the objective function is





















 

,

),

1

2

(

,

),

2

(

~

) ( ) ( 3

odd

is

r

r

n

T

even

is

r

r

n

T

F

K k s k K k s k (11) where

}.

1

2

,

2

{

,

)

1

(

)

(

1 ) ( ) (



















r

n

r

n

l

a

l

T

l j k j j k s k



Let

M

be odd. In this case

























 

,

),

2

(

,

),

1

2

(

) ( ) ( ) (

odd

is

r

r

n

d

even

is

r

r

n

d

G

K k s k K k s k s (12) and there exists a

(

2

n



r

)

-dimensional vector

w

such that (10) holds, and the optimum value of the objective function is





















 

.

),

2

(

,

),

1

2

(

~

) ( ) ( 3

odd

is

r

r

n

T

even

is

r

r

n

T

F

K k s k K k s k (13)

4. The parametric optimization problem

Now we can formulate the parametric optimization prob-lem. Recall that the following conditions hold:

If

M

is even, then the parametric optimization problem which has to be solved to obtain the solution of problem

P

is the following

P

_

:

maximize (1) subject to

(3),(14),(4)-(6),(8)-(10), where

F

~

₁(i) is (2)



i



I

,

F

~

₂ is (7),

F

~

₃ is (11).

If

M

is odd, then the parametric optimization prob-lem to be considered becomes

P

_

:

maximize (1) subject to (3),(14),(4)-(6),(8),(12),

(10), where

F

~

₁(i)



i



I

is (2),

F

~

₂ is (7),

F

~

₃ is (13).

Both

P

_ and

P

_ are nonlinear programming prob-lems with

3

n

 r

2



1

parameters

m

~

,

q

,

A

~

,

A

, with

d

r

n

 )

2



3

(

switching time variables ₁( )

,

,

n(k)r

k 





_

,

,

,

,

,

2( ) ) ( 1

k

K

k r n k









_

with

(

n



r

)

-dimensional

var-iable vector

v

and

(

2

n



r

)

-dimensional variable vec-tor

w

, with

(

3

n



2

r

)



(

d



1

)

nonlinear equality con-straints and some box concon-straints. The objective functions of the parametric problems are rather simple, but the equality constraints are nonlinear and cannot be solved explicitly. Remark that the constraints of the parametric optimization problem are necessary and sufficient condi-tions for the feasibility of problem

P

.

Conclusions

We have proposed and investigated an optimal control model with several kinds of communications in order to determine the optimal expenditure rate for each segment and each retailer in a multi-segment market. Remark that if there are only one retailer and one market segment (i.e.

1



 r

n

) and if we consider the whole communication

expenditure rate a priori subdivided into two parts, one for the retailer and one for consumers, we obtain the model presented in [1]. Moreover, for this case we have seen in particular that the sign of eigenvalues of the ma-trix of motion equations determines the type of optimal alternate communication. This implies that if saturation aversion and goodwill decay, which can be considered as negative factors for the firm, are “stronger” than word-of-mouth and goodwill productivity, which are positive for the firm, than it is more convenient to advertise only in the middle of the selling period. Otherwise it is conven-ient to advertise at the beginning of the selling period and then to refresh the goodwill of the firm at the end of the period. In the model considered here, the type of optimal alternate communication in the selling period depends from the evenness/oddness of the number of negative eigenvalues of the matrix of motion equations.

We hope that a similar kind of interpretation (the relation between positive and negative factors for the firm, the retailers and the market segments) can be found for this general case of the model. At this aim the investi-gation of the parametric optimization problem

P

seems to be particularly relevant.

References

[1] Bykadorov, I., Ellero, A. and Moretti, E.: “Minimi-zation of communication expenditure for seasonal Products”, RAIRO Operations Research, Vol.36, p. 109-127, 2002.

[2] Bykadorov, I., Ellero, A. and Moretti, E.: “A model for the marketing of a seasonal product with differ-ent goodwills for consumer and retailer”, Journal of Statistics & Management Systems, Vol. 6, p. 115-133, 2003.

[3] Favaretto, D. and Viscolani, B.: “A single season production and advertising control problem with bounded final goodwill”, Journal of Information and Optimization Sciences, Vol. 21, p. 337-357, 2000. [4] Hernández-Castaneda S., Zapato-Lillo P.: “Normal

form of some infinite extensive games and continui-ty properties”, to appear in International Journal of Biomedical Soft Computing and Human Sciences. [5] Hernández-Castaneda S., Zapato-Lillo P.:

“Existance of Nash equilibrium in infinite extensive games”, to appear in International Journal of Bio-medical Soft Computing and Human Sciences. [6] Kaku I., Li Zh. and Xu Ch.: “An Effective

Heuris-tic Algorithm based on Segmentation for Solving a Multilevel Lot-sizing Problem”, International Jour-nal of Biomedical Soft Computing and Human Sci-ences. V. 15, No. 1, Special Issue on Total Opera-tions Management, p. 53 - 59, 2010.

[7] Seierstad, A. and Sydsæter, K.: “Optimal Control Theory with Economic Applications”, North--Holland, Amsterdam, 1987.

Igor BYKADOROV He received the

Degree in Mathematics from Novosibirsk State University in 1983, Ph.D. degree in Mathematics at Sobolev Institute of Math-ematics, Novosibirsk. His present research interests include mathematical models for economic applications, game theory, pric-ing.

Andrea ELLERO He received the Degree

in Mathematics from the University of Padova, Ph.D. degree in Mathematics applied to Economics from University of Trieste. He currently investigates the ap-plication of mathematical models to mar-keting and revenue optimization serving as an associate professor.

Elena MORETTI She received the

Degree in Physics at the University of Padova. Here main research interests are optimal control and mathematical pro-gramming applied to management and