A Marketing-Oriented Inventory Model with Three-Component Demand Rate and Time-Dependent Partial Backlogging

International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-3, Mar- 2017]
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A Marketing-Oriented Inventory Model with
Three-Component Demand Rate and Time-
Dependent Partial Backlogging
Dr. Samiran Senapati
Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, West Bengal, India.
Abstract— This paper, an attempt has been made to
extend the model of “An EOQ model for perishable items
under stock-dependent selling rate and time-dependent
partial backlogging” with a view to making the model
more flexible, realistic and applicable in practice. Here,
objectives are to maximize the profit and minimize the
total shortage cost. In this model, fuzzy goals are used by
linear membership functions and after fuzzification, it is
solved by weighted fuzzy non-linear programming
technique. The model is illustrated with a numerical
example adopted partially from “An EOQ model for
perishable items under stock-dependent selling rate and
time-dependent partial backlogging”.
Keywords— EOQ; Perishable items; Partial back
logging; Fuzzification; Membership function.
I. INTRODUCTION
In the competitive market situation, it is commonly
observed that an increase in shelf space and glamorous
display for an item induce more consumers to buy it.
Recently, Dye and Ouyang (2005) investigated an
economic order quantity (EOQ) model for perishable
items under stock-dependent selling rate and time-
dependent partial backlogging. In two-component
demand, it is assumed that the demand rate is stock-
dependent down to a certain level and then it becomes
constant. But, it is commonly observed that the demand
rate will not be dependent on displayed stock level for a
huge amount of stock as all available stock cannot be
displayed properly and glamorously because of cost of
modern light, electronic arrangement and space will be
increased ( e.g. fashionable goods shop). It will be
dependent on displayed stock level within a range and
beyond this range, it will be quite uniform. This type of
demand rate is called three-component demand rate.
It has been recognized that one’s ability to make precise
statement concerning an inventory model diminishes with
increasing complexities of the system. Generally, it may
not be possible to define the objective goals precisely. In
reality, management is most likely to be uncertain of the
true value of parameters and due to many unforeseen
incidents like strike, hike in wages, increased
transportation cost etc; hence during the course of
business, a vendor or decision maker is forced to settle
down with a lower profit amount compared to the profit
as he/she normally has targeted due to adverse situation.
Moreover, shortages bring loss of goodwill for the
vendor. This loss can not be measured numerically. For
this reason, it is advisable to restrict the shortages as
much as possible to minimize the loss of goodwill. From
the above discussion, we may conclude that it is difficult
to determine the exact amount of profit and shortage cost
rather a range may be fixed for these. Hence, under these
phenomena the inventory model may be better treated in
a fuzzy system.
II. NOTATIONS AND MODELING
ASSUMPTIONS
In this section, we give the notations and assumptions
used throughout this chapter.
2.1 The inventory system involves only one item.
2.2 Replenishment rate is infinite and lead time is
zero.
2.3 θ, constant rate of deterioration. I(t) is the
inventory level at time t (Fig. 1).
2.4 p, the selling price per unit and A, the ordering
cost per order, are constant.
2.5 The unit cost C and the inventory carrying cost
as fraction i, per unit per unit time, are
constant.
2.6 Shortages are allowed and backlogged rate is
defined to be 1/[1+ δ(T-t)]. The backlogging
parameter δ is a positive constant. Shortage
cost is C2 per unit per unit time and R is the
fixed opportunity cost of lost sales per unit.
2.7 The demand rate D(p, I(t)), is dependent on
selling price and displayed stock level in the
show room with-in the stock level S0 to S1 and
beyond this range, it becomes constant with
respect to the display stock level. The
functional D(p, I(t)), is given by:

α (p) + βS1 I(t) ≥ S1,
α (p) + βI(t) S0 ≤ I(t) ≤ S1,
D(p, I(t)) = α (p) 0 ≤ I(t) ≤ S0,
 
 tTδ1
pα

I(t) ≤ 0
Where, β is a non-negative constant. α (p) is a non-
negative function of selling price p.
2.8 Shortages are allowed and backlogged rate is
defined to be 1/[1+ δ(T-t)]. The backlogging
parameter δ is a positive constant. Shortage
cost is C2 per unit per unit time and R is the
fixed opportunity cost of lost sales per unit.
2.9 T is the cycle time.
2.10TP and SC respectively denote the total
III. THE MATHEMATICAL MODEL
At the beginning of the order cycle the inventory level is
raised to Q afterwards as time progresses it is depleted by
combined effects of the demand and deterioration. The
pictorial representation of the inventory system is given in
Fig. 1. Therefore, the differential equations governing the
system during the period (0 ≤ t ≤ T) can be written as:
  1βSpαθI(t)
dt
dI(t)
 ,
I(t) ≥ S1,0≤ t≤t1 (1)
    tβIpαθI(t)
dt
dI(t)
1 ,
S0 ≤ I(t) ≤ S1, t1≤ t≤t2 (2)
 pαθI(t)
dt
dI(t)
 ,
0 ≤ I(t) ≤ S0, t2≤ t≤t3 (3)
 
 tTδ1
pα
dt
dI(t)

 ,
I(t) ≤ 0, t3≤ t≤T (4)
The solutions of the above differential equations, after
applying boundary conditions I(t1) = S0, I(t2) = S1, I(t3) =
0, are
  

























1
t1tθ
e
θ
1βSpαt1tθ
e1SI(t)
0≤ t ≤ t1 (5)
    
 
   1e
βθ
pα
eSI(t) ttβθttβθ
0
22


 
,
t1≤ t ≤ t2 (6) (6)
    1e
θ
pα
I(t) ttθ 3
 
,
t2≤ t≤t3 (7)
      tTδ1lntTδ1ln
δ
pα
I(t) 3 
t3≤ t≤T (8)
Ordering cost per cycle = A
Holding cost per cycle =
     








 
3
2
2
1
1 t
t
t
t
t
0
dttIdttIdttICi
Shortage cost per cycle (SC) =
       
T
t
32
3
dttTδ1lntTδ1ln
δ
pα
C
Opportunity cost due to lost sales per cycle =
 
 








T
t3
dt
tTδ1
1
1Rpα
Purchase cost
   
    








 1e
θ
βSpα
eStTδ1ln
δ
pα
C 11 θt1θt
13
Sales revenue per cycle =
       




 
2
1
1 t
t
t
0
1 dttβIpαdtβSpαS
    
  




 
T
t 3
t
t
1
3
3
2
dt
tTδ1
pα
dtβSpα
on integration and simplification of the relevant costs
mentioned above, the total profit per unit time TP
becomes,
TP=
       


 RδC
δ
pα
tTδ1lnC-p
δ
pα
223
    33 tTδ1ln-t-Tδ 
      
1eCiβS 12 ttβθ
 
 
    113
θt1
2
0
tβStpαp1e
θ
CiS
βθ
pα
βθ
S 1












         


 
2
1
23
ttθ
2
θ
βSpαCi
1ttθe
θ
pα
Ci 13
 1θte 1
θt1

  
 
 21 tt
βθ
Cipβpα
A 



 
  T1e
θ
βSpα
eSC
θt1θt
1
1











 , (9)
and total shortage cost per unit time,
       TdttTδ1lntTδ1ln
δ
pα
CSC
T
t
32
3
 
(10)
where S0 and S1 is given by,
 
 
 
 
  12 ttβθ
01 e
βθ
pα
S
βθ
pα
S 









 , (11)
     23 ttθ
0 e
θ
pα
θ
pα
S 
 , (12)
Now from (11) and (12) we get,

 
 
 
 
 

















βθ
pα
S
βθ
pα
S
ln
βθ
1
tt
0
1
12 , (13)
 
 














θ
pα
θ
pα
S
ln
1
tt
0
23

, (14)
The above two equations implies
t2 – t1> 0, (15)
and t3 – t2> 0, (16)
and the initial lot size
  








 11θt
e
θ
1βSpα
1θt
e1SI(0)Q ,
(17)
Replacing t1by 0 first, substitute t2 and t3 by t1, we can
observed that the above profit function will be same as
the profit function of Dye and Ouyang (2005).
5.1. Crips model
In crisp environment multi-objective problem of
maximizing total profit and minimizing the total shortage
cost can be written as follows:
Max TP
Min SC
Subject to,
t2 – t1> 0
t3 – t2>0
where t1, t2, t3, T ≥ 0. (18)
5.2Fuzzy model
Since seller’s maximum average revenue and minimum
total shortage cost per unit time becomes imprecise in
nature, the above model in fuzzy sense can be
represented as:
TPx~Ma
SCn~Mi
Subject to,
t2 – t1> 0
t3 – t2>0
where t1, t2, t3, T ≥ 0. (19)
5.3 Fuzzy goal programming of model
The fuzzy multi-objective problem can be formulated as
a FNLGP as follows:
Find (t1, t2, t3, T)T
subject to the constraints
f1( t1, t2, t3, T ) = -TP ≤ -f01
f2( t3, T ) = SC ≤ f02
t2 – t1> 0
t3 – t2>0
where t1, t2, t3, T ≥ 0.
Here, the fuzzy goal of objectives, i.e. total average profit
and total shortage cost, are (f01-P01, f01) and (f02, f02+P02)
respectively, and there linear MFs are consider as
follows:
0,for f1(t1, t2, t3, T) ≤ - f01+P01
µ1(f1(t1, t2, t3, T)) =
 
01
013211
P
fT,t,t,tf
1

 ,
for -f01 ≤ f1(t1, t2, t3,T) ≤ -f01+P01
1, for f1(t1, t2, t3, T) ≤ - f01
i.e.
0, for TP ≤ f01- P01
µ1(TP) =
01
01
P
f-TP
1 , for f01- P01≤ TP ≤ f01
1,for TP ≥ f01
and
0, for SC ≥ f02 + P02
µ2(SC)=
02
02
P
fSC
1

 ,for f02≤ SC ≤f02 +
P02
1, for SC ≤ f01
Using the weights to represent different importance for
the objectives, the problem can be written as follows:
Max F = w1µ1(TP) + w2µ2(SC)
Subject to
f01-P01≤ µ1(TP) ≤ f01
f02≤ µ2(SC) ≤ f02+P02
t2 – t1> 0
t3 – t2>0
w1 + w2 = 1
where t1, t2, t3, T ≥ 0.
IV. NUMERICAL EXAMPLES
To illustrate the above inventory models, values of
the system parameters are considered as:
A = 250.0, β = 0.3, θ = 0.08, C = 5.0, i = 0.35, C2=3.0,
p=7.0, R=5.0, δ = 10,   r
K(p)pα 
 , K = 20000.0,
p=7.0,S0 = 100.0, S1= 300.0, r = 1.5, f01 = -$750.0, f02 =
$30.0, P01 = -$625.0, P02 = $20.0.
The optimal values of t1, t2, t3, t4 along with total profit,
total shortage cost and lot-size are displayed below:
From Table-1 and Table-2, it is observed that when a
seller takes care of his profit only, the seller makes
maximum revenue at the cost of his reputation and
goodwill. Similarly when the seller only takes care of his
shortage cost, his total revenue is lower. As expected,
when interests of both seller’s total revenue and shortage
cost are considered, then total revenue and shortage costs
become moderate, i.e. it lies between the above
mentioned levels.

V. FIGURES AND TABLES
Table.1: Results for Crisp model
Crisp model Equal
weight for
profit &
shortage
cost
First priority
for profit
First
priority
for
shortage
cost
t1 0.2489 0.2531 0.2613
t2 0.5214 0.5310 0.5124
t3 0.7025 0.7112 0.7124
T 0.9678 0.9852 0.9675
Profit ($) 709.62 717.39 693.71
Shortage cost
($)
58.29 63.69 54.20
Lot-size 537.81 542.05 550.39
Table.2: Results for Fuzzy model
Fuzzy model Equal
weight for
profit &
shortage
cost
First priority
for profit
First
priority
for
shortage
cost
t1 0.2521 0.2641 0.3196
t2 0.5247 0.5482 0.5772
t3 0.7288 0.7441 0.7738
T 0.9558 0.9885 0.9435
Profit ($) 819.51 825.38 744.12
Shortage cost
($)
43.59 50.04 25.17
Lot-size 543.05 533.22 610.36
VI. CONCLUSION
A multi-objective inventory model of deteriorating item
with stock and price dependent demand, with shortages is
developed. Here a real-life inventory problem faced by
the inventory practitioners is considered. The purpose of
this chapter is to investigate an inventory model for
deteriorating item with three-component demand rate;
permitting shortage and time-proportional backlogging
rate within the economic order quantity (EOQ)
framework. In the existing model Dye and Ouyang
(2005), authors considered the demand rate dependent on
the current displayed stock, i.e. the demand rate will be
high and high for more and more displayed stock in the
showroom. This is somehow unrealistic. The stock
dependency nature must occur within a range, and
beyond this range it will be quite uniform. Selling price is
also an influencing factor on demand. Under fire over
various financial ethical issues globally, some attention
must be need to the replenishment cost so that it becomes
minimum along with the maximum profit. Such a
realistic problem has been modeled and solved under
crisp and fuzzy environment. Since the proposed model
has been formulated with imprecise informations, the
decision maker may choose that solution which suits
him/her best respect to conditions and restrictions. Till
now, only a very few researchers have considered such a
realistic phenomenon, though several papers dealing with
an EOQ model with deterioration and time-dependent
partial backlogging are available.
The scope of application of the model in supermarkets is
open however, success depends on correctness of the
estimation of input parameters. To estimate the
parameters, demands of the same kind product in
different supermarkets have to be observed and analyzed
over long time.
ACKNOWLEDGEMENTS
I am grateful to express my reverence to my honorable
teachers of Department of Mathematics, University of
Kalyani. For the source of relevant of information, I am
indebted to the librarians of Indian Institute of
Management, Kolkata, and Indian Statistical Institute,
Kolkata. I am also thankful to the librarians and the others
members of staff of the libraries of the department.
I convey my heart left thanks to Dr. Shibaji Panda,
Bengal Institute of Technology, Kolkata – 700150, Dr.
Subrata Saha, Dr. Kanailal Banerjee and all of my friends,
Colleagues of Nabadwip Vidyasagar College, Research
Scholars and all other well-wishers of Department of
Mathematics, University of Kalyani for the co-operation,
helps and inspiration during the period of my research
work.
I sincerely remember and acknowledge the
encouragement of my family members extended to me to
complete my research work.
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