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Integrating marketing and production decisions
Engineering
Costs and Production Economics,
Elsevier Science Publishers
B.V., Amsterdam
INTEGRATING
15 (1988) 387-390 - Printed in Hungary
387
MARKETING AND DECISIONS R. McTavish Concordia
PRODUCTION
and S. K. Goyal
University,
Montreal.
Canada
INTRODUCTION
NOTATIONS
The problem of determining economic production policy has been extensively discussed in the Management Science literature [see Peterson and Silver (1979) Starr and Miller (1967), Hadley and Whitin (1963)] under various conditions and assumptions. Recently researchers have shown an interest in the study of production-inventory systems under varying marketing conditions like the frequency of advertisements, the effect of price elasticity etc. Ladany and Sternlieb (1974) consider the effect of price elasticity in demand. Subramanyam and Kumaraswany (1981) consider the effect of price elasticity as well as the frequency of advertisements on the demand. However, they assume a linear relationship between the demand and the frequency of advertisements. In this paper, we develop an integrated net profit model incorporating price elasticity, frequency of advertisements and the batch quantity. We propose a fast converging iterative procedure for determining the economic operating policy. The net profit model is based on the following assumptions: 1. The demand for the product is assumed to be at a uniform rate over time. 2. The advertisements are placed at equal time intervals. 3. Shortages are not permitted. 4. The items produced in a batch are put into the inventory only at the completion of a batch. 5. Time horizon is infinite. 6. Maximization of the net profit per unit of time is the objective.
D= P s C= h A R e= N= Z=
= = = = =
Demand per unit of time Unit selling price Cost of manufacturing setup Unit production cost Unit stock holding cost per unit of time Cost of an advertisement Rate of production Production batch quantity Frequency of advertisement Net profit per unit of time
THE MATHEMATICAL
MODEL
The net profit per unit of time is given by the sales revenue per unit of time minus the total relevant costs per unit of time. The total relevant costs per unit of time consist of the production cost, the cost of advertisements, the cost of production setups and the stock holding cost. Therefore, the net profit is given by Z=DP-DC-AN-z-” e
.2
Note that in the above model DP DC AN
WQ
= sales revenue; = production cost; = cost of advertisements; = cost- of production setups; /2 = stock holding
cost.
and
388
On rearranging
the terms in the total cost model
we get
p-c-S-53 Q
>
- AN-
ZR
fh.
(1)
We modify Kotler’s demand model [Kotler (1971)] to take into account the effects of unit selling price and the frequency of advertisements on the demand for the product. The demand is estimated from the following model D = F(h+fNB) where f&O, On from
It may be pointed out that we cannot determine P from (4) without knowing Q which in turn cannot be determined from (5) without knowing P and N. For determining N from (6) we need to have prior knowledge of P and Q. Therefore, an iterative search procedure for determining the decision variables. In the ith iteration, we determine the value of the decision variables by solving the following equations: DimI =,P;-,(b+fN:_l)
Q)
(7)
(8)
e, h, f and g are constants. Note that b 80, e< - 1 and g< 1. substituting the value of the estimated demand (2) in (1) we get
(3) At a given value of Q and N, the net profit is maximized by choosing the value of P which satisfies the equation aZ/aP = 0. On differentiating (3) partially with respect to P and equating the first derivative equal to zero, we get P=
&
.+;+g. (
(4)
>
At a given value of P and N, the net profit is maximized by equating the first partial derivative of Z with respect to Q equal to zero. We get
Q=
Q = [h(::“1)]‘;2.
(5)
R Note that F(b+jiV) Similarly at the economic frequency N=
= D. given P and Q values, the of advertisements is given by
A
p_~-S-!z!! Q
2R
I/e1 1
(6)
The search procedure is started by setting i = 1 and with prior values of N, and P,. We start with N, = 0 and P, = eC/(l + e) and determine Qr from (8). P, is determined from (9) at Q = Qr. At P = P, and Q = Q,, the economic value of Iv’ = N, is obtained from (10). This completes the first iteration. Having obtained Q,, P, and N,, the second iteration is carried out, The iterative procedure is stopped in the kth iteration if P, = Pk 1 and Nk = N,_,. The economic value of the decision variables are given by Q* = Qlr, P* = P, and N* = Nk. During any iteration the demand and the net profit can be evaluated from (7) and (11) respectively. We will now illustrate the iterative procedure with the help of the following example.
AN EXAMPLE A C
= $1,000 for advertisement = $5 per item
389 h h
l/g-1
From (10)
= $1 per item per year = 1,000,000
1
;
N, =
I ;;,oOO = 0.6 = Pm2 (1,000,000+3,000 = 100,000 per year = $100 per setup
g
D R S
=
N”.6)
100,000x0.6x(10.16)~2
10.16-5(
In the first iteration
ec
PO = ~ I+e From (7) From (8)
l1
THE SOLUTION
0.6
100 1,348 x 1 _ __ 2 x 100,000 1,348
(i= I).
= 10 and N,=O.
- 1
= 14.98 per year.
From (1)
Do = 10,000 per year.
z,
=
P;(b+fNy) P,-c-g
I
(
_ AN _ 1
-
- g >
Qlh_ 2R
= (10.16))2(1,000,000+
x(14.98)O.“)
10.16-5-
100,000x
+$
-
( = 1,348 per batch.
_
From (9) p, =
.
1348x 1 2 x 100,000 >
c
c+s+e,h Pi
l+e
- 1000 x 14.98 -
- ~- 1348 x 1 = $58,510.67. 2 x 100,000
=
2R >
100 __~ 1,348
Using P, and N, values, the second iteration performed. The results of the computations given in the Table.
= $10.16 per item.
TABLE Results of computations Values of decision
Prior values Iteration i
1 2 1 * q~timal
N,-, 10 10.16 IO.14
profit
0 14.98 15.05
Di-1
Q,
pi
-
N,
from (7)
from (8)
from (9)
from (IO)
10,000 14.602 14.674
I348 1596 1600
10.16 10.14 10.14
14.98 15.0s 15.05
& from (11) 585 10.67 58539.83 58539.84*
was are
390
CONCLUDING
REMARKS
The iterative procedure given in this paper converge:, very rapidly. Using the approach given in the paper, the ma1 ketin; an2 production decisions can be incorporated in a single decision model.
REFERENCES Hadley.
G. and
.Systm.s. Kotler. P.
Whitin,
T. M
(1963).
Anufysi.v of Inventory
Prentice-Hall, Inc., En&wood Cliffs, New Jersey. (I 91 I), Murkering Decision Making. A Model Building
Holt. Reinhart
Approach,
Ladany. S. and Sternlieb. quantities order;!:2
and Winston,
Inc.. New York.
A. (1974). “The interaction ofeconomic policies”. 4 IIE and marketing
‘l’RANiV.T.4CT‘IONS. 6,
I.
ACKNOWLEDGEMENT
‘This research was supported by grant No. A5004 of the Natural Sciences and Engineering Research Council of Canada.
Subramanyam, formula
under
E. S
and
varling
A/II: 7RANSAC’77ONS.
Kumaraswamy. rnarks.ling I.?. 4.
policies
S. (I9RI).
“EOQ
and conditions”.
Costs and Production Economics,
Elsevier Science Publishers
B.V., Amsterdam
INTEGRATING
15 (1988) 387-390 - Printed in Hungary
387
MARKETING AND DECISIONS R. McTavish Concordia
PRODUCTION
and S. K. Goyal
University,
Montreal.
Canada
INTRODUCTION
NOTATIONS
The problem of determining economic production policy has been extensively discussed in the Management Science literature [see Peterson and Silver (1979) Starr and Miller (1967), Hadley and Whitin (1963)] under various conditions and assumptions. Recently researchers have shown an interest in the study of production-inventory systems under varying marketing conditions like the frequency of advertisements, the effect of price elasticity etc. Ladany and Sternlieb (1974) consider the effect of price elasticity in demand. Subramanyam and Kumaraswany (1981) consider the effect of price elasticity as well as the frequency of advertisements on the demand. However, they assume a linear relationship between the demand and the frequency of advertisements. In this paper, we develop an integrated net profit model incorporating price elasticity, frequency of advertisements and the batch quantity. We propose a fast converging iterative procedure for determining the economic operating policy. The net profit model is based on the following assumptions: 1. The demand for the product is assumed to be at a uniform rate over time. 2. The advertisements are placed at equal time intervals. 3. Shortages are not permitted. 4. The items produced in a batch are put into the inventory only at the completion of a batch. 5. Time horizon is infinite. 6. Maximization of the net profit per unit of time is the objective.
D= P s C= h A R e= N= Z=
= = = = =
Demand per unit of time Unit selling price Cost of manufacturing setup Unit production cost Unit stock holding cost per unit of time Cost of an advertisement Rate of production Production batch quantity Frequency of advertisement Net profit per unit of time
THE MATHEMATICAL
MODEL
The net profit per unit of time is given by the sales revenue per unit of time minus the total relevant costs per unit of time. The total relevant costs per unit of time consist of the production cost, the cost of advertisements, the cost of production setups and the stock holding cost. Therefore, the net profit is given by Z=DP-DC-AN-z-” e
.2
Note that in the above model DP DC AN
WQ
= sales revenue; = production cost; = cost of advertisements; = cost- of production setups; /2 = stock holding
cost.
and
388
On rearranging
the terms in the total cost model
we get
p-c-S-53 Q
>
- AN-
ZR
fh.
(1)
We modify Kotler’s demand model [Kotler (1971)] to take into account the effects of unit selling price and the frequency of advertisements on the demand for the product. The demand is estimated from the following model D = F(h+fNB) where f&O, On from
It may be pointed out that we cannot determine P from (4) without knowing Q which in turn cannot be determined from (5) without knowing P and N. For determining N from (6) we need to have prior knowledge of P and Q. Therefore, an iterative search procedure for determining the decision variables. In the ith iteration, we determine the value of the decision variables by solving the following equations: DimI =,P;-,(b+fN:_l)
Q)
(7)
(8)
e, h, f and g are constants. Note that b 80, e< - 1 and g< 1. substituting the value of the estimated demand (2) in (1) we get
(3) At a given value of Q and N, the net profit is maximized by choosing the value of P which satisfies the equation aZ/aP = 0. On differentiating (3) partially with respect to P and equating the first derivative equal to zero, we get P=
&
.+;+g. (
(4)
>
At a given value of P and N, the net profit is maximized by equating the first partial derivative of Z with respect to Q equal to zero. We get
Q=
Q = [h(::“1)]‘;2.
(5)
R Note that F(b+jiV) Similarly at the economic frequency N=
= D. given P and Q values, the of advertisements is given by
A
p_~-S-!z!! Q
2R
I/e1 1
(6)
The search procedure is started by setting i = 1 and with prior values of N, and P,. We start with N, = 0 and P, = eC/(l + e) and determine Qr from (8). P, is determined from (9) at Q = Qr. At P = P, and Q = Q,, the economic value of Iv’ = N, is obtained from (10). This completes the first iteration. Having obtained Q,, P, and N,, the second iteration is carried out, The iterative procedure is stopped in the kth iteration if P, = Pk 1 and Nk = N,_,. The economic value of the decision variables are given by Q* = Qlr, P* = P, and N* = Nk. During any iteration the demand and the net profit can be evaluated from (7) and (11) respectively. We will now illustrate the iterative procedure with the help of the following example.
AN EXAMPLE A C
= $1,000 for advertisement = $5 per item
389 h h
l/g-1
From (10)
= $1 per item per year = 1,000,000
1
;
N, =
I ;;,oOO = 0.6 = Pm2 (1,000,000+3,000 = 100,000 per year = $100 per setup
g
D R S
=
N”.6)
100,000x0.6x(10.16)~2
10.16-5(
In the first iteration
ec
PO = ~ I+e From (7) From (8)
l1
THE SOLUTION
0.6
100 1,348 x 1 _ __ 2 x 100,000 1,348
(i= I).
= 10 and N,=O.
- 1
= 14.98 per year.
From (1)
Do = 10,000 per year.
z,
=
P;(b+fNy) P,-c-g
I
(
_ AN _ 1
-
- g >
Qlh_ 2R
= (10.16))2(1,000,000+
x(14.98)O.“)
10.16-5-
100,000x
+$
-
( = 1,348 per batch.
_
From (9) p, =
.
1348x 1 2 x 100,000 >
c
c+s+e,h Pi
l+e
- 1000 x 14.98 -
- ~- 1348 x 1 = $58,510.67. 2 x 100,000
=
2R >
100 __~ 1,348
Using P, and N, values, the second iteration performed. The results of the computations given in the Table.
= $10.16 per item.
TABLE Results of computations Values of decision
Prior values Iteration i
1 2 1 * q~timal
N,-, 10 10.16 IO.14
profit
0 14.98 15.05
Di-1
Q,
pi
-
N,
from (7)
from (8)
from (9)
from (IO)
10,000 14.602 14.674
I348 1596 1600
10.16 10.14 10.14
14.98 15.0s 15.05
& from (11) 585 10.67 58539.83 58539.84*
was are
390
CONCLUDING
REMARKS
The iterative procedure given in this paper converge:, very rapidly. Using the approach given in the paper, the ma1 ketin; an2 production decisions can be incorporated in a single decision model.
REFERENCES Hadley.
G. and
.Systm.s. Kotler. P.
Whitin,
T. M
(1963).
Anufysi.v of Inventory
Prentice-Hall, Inc., En&wood Cliffs, New Jersey. (I 91 I), Murkering Decision Making. A Model Building
Holt. Reinhart
Approach,
Ladany. S. and Sternlieb. quantities order;!:2
and Winston,
Inc.. New York.
A. (1974). “The interaction ofeconomic policies”. 4 IIE and marketing
‘l’RANiV.T.4CT‘IONS. 6,
I.
ACKNOWLEDGEMENT
‘This research was supported by grant No. A5004 of the Natural Sciences and Engineering Research Council of Canada.
Subramanyam, formula
under
E. S
and
varling
A/II: 7RANSAC’77ONS.
Kumaraswamy. rnarks.ling I.?. 4.
policies
S. (I9RI).
“EOQ
and conditions”.