Abhishek Kanti Biswas
R. K. M. Vivekananda Centenary College, India
E-mail: a.k.biswas@hotmail.com
Sahidul Islam
University of Kalyani, India
E-mail: sahidul.math@gmail.com
Submission: 04/09/2018
Revision: 18/09/2018
Accept: 15/10/2018
ABSTRACT
For any business, the inventory
system takes a monumental part. Keep this aspect in mind, we formulate
multi-objective displayed EPQ model consider with non-instantaneous
deteriorating items where production depends on demand and variable demand
pattern depends on display shelf-space, selling price and frequency of advertisement
of the items. The customers are more attracted to buy an item by observing
display shelf-space, selling price and advertisement. For any inventory system,
defective and deteriorated items are now and again come back to providers for a
discount or credit. Here price discount is available for deteriorated and
defective items. Holding cost varies with time where shortages are allowed and
fully backlogged. Delay in payment is permissible. Fuzzy environment touches
the reality instead of the crisp environment. So, we assumed the cost
components (parameters) as Linear Fuzzy Numbers and then using Nearest Interval
Approximation Method the parameters converted to parametric interim -valued
function for solving the model in a fuzzy environment. Finally, several
numerical examples are given to illustrate the model. Also graphs are given for
the optimal values of
Keywords: EPQ inventory, Demand dependent production, Fully backlogged,
Triangular Fuzzy Numbers, Nearest Interval Approximation Method.
1. INTRODUCTION
To controlling any business,
Inventory analysis is an important part. Analysts around there have broadened
examination concerning different inventory models with contemplations of demand
designs, deterioration, deficiencies, payment options, arrange cycles and their
combinations. Economic Production Quantity Inventory Model requires watchful
and mindful methodology.
By controlling quality,
manufacturers ordinarily need to deliver the perfect volume of items. In the
current years, the deteriorating items have gotten much consideration in light
of the fact that the vast majority of the physical merchandise experience decay
or deterioration after lifetime, a case being natural products, vegetables,
unstable products et cetera.
In such circumstance price discount
are normal practices by a provider to encourage the client to buy a
considerable measure estimate or defective items other than standard buy. In
1963, the first attempt at the analysis of the deteriorating inventory model
was made by Ghare and Schrader (1963) with a constant rate of
decay.
Wee (1999) developed a deteriorating
inventory model with a quantity discount, pricing, and partial back ordering.
Connecting above, Ouyang et. al. (2006), researched an inventory model for
non-instantaneous deteriorating items with the permissible delay in payments.
Holding costs are one component of total inventory costs that are associated
with storing inventory that remains unsold. Also, Demand is different for
different types of item, item quality, customers of the certain area, selling
price, advertisement of the items and different period of time.
According to retailers’ reputation
and quality of items, in some inventory system, customers are waiting for the
next replenishment. So, fully backlogged is acceptable. Mondal and Islam (2016),
examined a fuzzy EOQ display for deteriorating things, with constant demand,
shortages and fully backlogging. While modelling an inventory problem, display
inventory level demands have an effect on sales for many retail products.
According to Whitin (1957), “For
retail stores, the inventory control problem for style goods is further
complicated with the fact that inventory and sales are not independent of one
another. An increase in inventory may bring about increased sales of some
items”. In the present competitive market, the effect of marketing policies and
conditions such as the price variations and the advertisement of an item change
its demand pattern amongst the public.
Very few OR researchers and
practitioners studied the effects of price variations and the advertisement on
the demand rate for items.
Ladany and Sternleib (1974)
studied the effect of price variation on selling and consequently on EOQ.
Nonetheless, they didn't think about the impact of the advertisement.
Subramanyam and Kumaraswamy (1981) and Urban (1992), created inventory models
consolidating the impacts of selling price an advertisement of a thing.
Sometimes, the supplier will offer the retailer a trade credit period in a
competitive market environment, in paying for the amount of purchasing cost.
Usually, if the retailer sells the
whole items inside the cut-off period and settled the outstanding amount within
the permitted fixed settlement period there is no charge. If the settled pay
date passes, interest is changed by a supplier for the rest of the products
that are in stock. Numerous specialists have thought about the aforesaid
condition in various ways. Ou (2016), built up a model of an optimal
replenishment policy under Conditions of permissible delay in payment.
Applying the conventional inventory
model as they are generally, leads to erroneous decisions, are not capable of
representing real-life situations. Fuzzy inventory models fulfil that gap.
Different fuzzy inventory models occur due to fuzzy various cost parameters in
the total cost. Fuzzy set theory, introduced by Zadeh (1965), has been
receiving considerable attention from researchers in production and inventory
management as well as in other fields.
Zadeh and Bellmann (1970) proposed a
mathematical model for basic leadership in a fuzzy domain. Dubois and Prade (1978)
characterizes a few activities on fuzzy numbers. Zimmermann (1985) made an
attempt to use the fuzzy sets in operation research. Grzegorzewski (2002)
investigate the nearest interval approximation of a fuzzy number. Mondal and
Roy (2006), approximated Linear Fuzzy Number in a fuzzy environment.
In the present work, we developed a
multi-objective fuzzy deterministic inventory model for non-instantaneous
deteriorating items where demand depends on display shelf-space, selling price
and advertisement. Holding cost is expressed as the function of time. Shortages
are allowed and fully backlogged. The inventory costs are taken as Linear Fuzzy
Number. Using Nearest Interval Approximation Method, the parameters converted
to parametric interim -valued function for solving the model in a fuzzy
environment. The solution for minimizing the total cost has been derived. To
the author’s best of knowledge such type of model has not yet been discussed in
the inventory literature.
2. DEFINITIONS AND FUZZY PRELIMINARIES
For this model, we need the following definitions:
·
Definition 2.1: A fuzzy
set
·
Definition 2.2: A fuzzy set
·
Definition 2.3: A fuzzy set
·
Definition 2.4: The
·
Definition 2.5: A fuzzy number is a fuzzy set in the universe of
discourse X that is both convex and normal. The following figure 1
shows a fuzzy number
Figure 1: Fuzzy Number
Above figure-1 shows a fuzzy number
·
Definition 2.6: Among the
various shapes of fuzzy number, triangular fuzzy number (TFN) is the most
popular one.
Figure 2: Triangular Fuzzy Number Ã(a1,
a2, a3)
·
Definition 2.7: An interval
number A is defined by an order pair of real numbers as follows:
·
Definition 2.8: Here, we will propose interval approximation operator called the
nearest interval approximation which approximates a fuzzy number by a crisp
model. Suppose,
Given
with
respect to
In order to
minimize
And
and then to
solve
The
solution is
Moreover,
since
So,
Therefore,
the interval
·
Definition 2.9: Let,
By the nearest interval approximation method the lower limit of the
interval is
and the upper limit of the interval is
Therefore, the interval number considering
·
Definition 2.10: Let [
3. Mathematical Model
This inventory model is developed on the basis of the
following Assumptions and Notations which are used throughout this paper in
Crisp and Fuzzy Environment.
3.1.
Notations:
·
·
·
·
·
·
·
·
·
Q ( =
·
·
·
·
·
·
·
·
3.2.
Assumptions:
·
The demand rate
·
Production rate
·
Holding cost is
·
The horizontal planning
takes place at an infinite rate.
·
Shortages are allowed and fully backlogged.
·
Lead time is zero or negligible.
·
There is no replenishment
or repair of deteriorating and defective items takes place in the given cycle.
3.3.
Mathematical
formulation
Let, the manufacturer start to
produce items to satisfy the arriving demands in the inventory system at
the beginning of each cycle when t = 0. At end
of
Figure 3: Graphical Representation of Inventory System
According
to figure-3, the inventory partitions as follows:
a) The Inventory Level in
With the initial Condition
And
b) The Inventory
Level in
With the help of the Conditions
And
c) The Inventory Level in
With the Conditions
And
d) The Inventory Level in
With the boundary Conditions
And
Also, we have
Thus the order size during total time interval [ 0,
Q =
According to above
discussion, the following cost function can be derived.
1
The set-up cost
during the cycle:
2 The production cost during the cycle:
3 The Inventory Holding cost
during the period [ 0
4 The Deteriorating cost during the period [
5
The shortage cost per cycle :
6 The advertisement cost
during the cycle:
7 The price discount during
the period [
Now, we will discuss
the following two cases:
Case-1: When
For 0
Beyond the fixed
settlement period, the unsold stock is financed with an interest rate
According to aforesaid definition of
total cost, the total average net cost function as follows:
Subject to,
Therefore, using equation (3.4.13), the equations (3.4.24)
reduced to
Subject to,
Now, the necessary condition for the
total average cost of the system is minimize if equation (3.4.25) is satisfy,
And
The solution, which may be called feasible
solution of the problem, of the equations (3.4.26) and (3.4.27) give the
optimal solutions
And
However, it’s difficult to solve the
problem by deriving an explicit equation of the solutions from equations (3.4.26) and (3.4.27). Therefore, we
solve the optimal service level
Case-2: When
For
Figure
5: The behavior of inventory model is demonstrated
According to aforesaid definition of
total cost, the total average net cost function as follows:
Subject to,
Therefore, using equation (3.4.13), the equations
(3.4.31) reduced to
Subject to,
Similarly, the necessary condition
for the total average cost of the system is minimize if equation (3.4.32) is satisfy the
conditions stated above equations(3.4.26) and (3.4.27)under certain sufficient conditions ( i.e.
inequalities (3.4.28) and (3.4.29)).
3.4.
Fuzzy
Production Inventory Model
For Case-1:
In our inventory model, we have
considered that the cost parameters
We now form interval numbers for
linear fuzzy parameters with the help of
the procedureof the nearest interval approximation of a fuzzy number stated
above i.e., [
Subject to,
Now, the necessary condition for the
total average cost of the system is minimize if equation (3.4.33) is satisfy,
And
The solution, which may be called
feasible solution of the problem, of the equations (3.4.34) and (3.4.35) give the optimal
solutions
And
However, it’s difficult to solve the
problem by deriving an explicit equation of the solutions from equations (3.4.34) and (3.4.35). Therefore, we
solve the optimal service level
For Case-2:
In our inventory model, we have
considered that the cost parameters
We now form interval numbers for
triangular fuzzy parameters with the help of the procedure of the nearest
interval approximation of a fuzzy number stated above i.e., [
Subject to,
Similarly, the necessary condition
for the total average cost of the system is minimize if equation (3.4.37) is satisfy the
conditions stated above equations (3.4.34) and (3.4.35)under certain sufficient conditions ( i.e.
inequalities (3.4.36) and (3.4.37)).
4. Numerical Solution
4.1.
Solution
in Crisp Environment:
A manufacturing company produces and sell of items. The
necessary information’s for the concerned item is given as follows:
4.1.1. Case-1,
Example-1:
From the past records it is seen that
display shelf-space
(s) = 35sq. inch per unit item,
frequency of advertisement
(f) = 5per cycle,
power of advertisement (r) = 3per cycle,
holding cost exponential parameter (
selling price (p) =
$ 25per unit of an item,
the constant (k) = 1.4proportional to demand per unit per cycle
defective units (u)=26 per cycle.
the fixed set up cost (
the constant (l) = 0.12 proportional to
selling price per unit per cycle,
the storage cost (
the advertisement cost (
the discount (d) = $ 0.54per unit per cycle.
the delay in
payment time (M)=1.2 days,
total amount of
capital investment (
Using the solution
procedure describe above, the result are presented in the following table:
Table
1: Optimal solution of the proposed model in Crisp Environment
|
|
|
0.9149060 |
0.2994649E-01 |
693.4914 |
4.1.2. Case-1,
Example-2:
From the past records it is seen that for any
company-
display shelf-space
(s) = 0.3 sq. feet per unit item,
frequency of
advertisement (f) = 5 per cycle,
power of advertisement (r) = 3.5 per cycle,
holding cost exponential parameter (
selling price (p) =
$ 41 per unit of an item,
the constant (k) = 1.25 proportional to demand per unit per cycle,
defective units (u) = 6 per cycle.
the fixed set up cost (
the constant (l) = 0.4 proportional to
selling price per unit per cycle,
the storage cost (
the advertisement cost (
the discount (d) = $ 2 per unit item per
cycle,
the delay in
payment time (M)=1.2 days
Total amount of
capital investment (
Using the solution
procedure describe above, the result are presented in the following table:
Table
2: Optimal solution of the proposed model in Crisp Environment
|
|
|
6.598888 |
0.7749249 |
268.0467 |
4.1.3. Case-2,
Example -1:
From the past records it is seen that
display shelf-space
(s) = 9 sq. inch per unit item,
frequency of
advertisement (f) = 4 per cycle,
power of advertisement (r) = 3.4 per cycle,
holding cost exponential parameter (
selling price (p) =
$ 16 per unit of an item,
the constant (k) = 1.28 proportional to demand per unit per cycle
defective units (u) = 5 per cycle.
the fixed set up cost (
the constant (l) = 0.3 proportional to
selling price per unit per cycle,
the storage cost (
the advertisement cost (
the discount (d) = $ 0.2 per unit per cycle.
total amount of
capital investment (
Using the solution
procedure describe above, the result are presented in the following table:
Table
3: Optimal solution of the proposed model in Crisp Environment
|
|
|
1.353385 |
0.7308970E-01 |
312.9635 |
4.1.4. Case-2,
Example - 2:
From the past records it is seen that
display shelf-space
(s) = 0.3 sq. feet per unit item,
frequency of
advertisement (f) = 5 per cycle,
power of advertisement (r) = 4 per cycle,
holding cost exponential parameter (
selling price (p) =
$ 65 per unit of an item,
the constant (k) = 1.3 proportional to demand per unit per cycle,
defective units
(u) = 6 per cycle,
the fixed set up cost (
the constant (l) = 0.2 proportional to
selling price per unit per cycle,
the storage cost (
the advertisement cost (
the discount (d) = $ 0.9 per unit per cycle,
total amount of
capital investment (
Using the solution procedure
describe above, the result are presented in the following table:
Table
4: Optimal solution of the proposed model in Crisp Environment
|
|
|
6.865024 |
0.2645908 |
230.4581 |
4.2.
Solution
in Fuzzy Environment
4.2.1. Case-1,
Example-1:
From
the above example in 4.1.1,If we take the input costs, total capital investment and selling price
of the proposed inventory model as linear
fuzzy number, then the fuzzy numbers are
p = [23, 27] ⇒
The optimal solution of the fuzzy
model by interval-valued parametric geometric programming is presented in Table
below:
Table
5: The Optimal Solution for proposed Fuzzy Inventory Model
|
|
|
|
0.0 |
0.8395553 |
0.2952033E-01 |
689.9308 |
0.2 |
0.8675971 |
0.2967504E-01 |
691.2982 |
0.4 |
0.8965497 |
0.2983279E-01 |
692.6211 |
0.5 |
0.9113762 |
0.2991285E-01 |
693.2643 |
0.6 |
0.9264409 |
0.2999371E-01 |
693.8944 |
0.8 |
0.9572987 |
0.3015788E-01 |
695.1126 |
1.0 |
0.9891514 |
0.3032538E-01 |
696.2696 |
Here we have given graphs, which
shown how change the value of
|
|
Graph 1: |
Graph 2: |
|
|
Graphic 3:
4.2.2. Case-1,
Example-2:
From
the above example in 4.1.1.2, If we take the input costs’, total capital investment and selling price
of the proposed inventory model as linear fuzzy number, then the fuzzy numbers are
p = [38, 44] ⇒
The optimal solution of the fuzzy
model by interval-valued parametric geometric programming is presented in Table
below:
Table
6: The Optimal Solution for Proposed Fuzzy Inventory Model
|
|
|
|
0.0 |
5.889006 |
0.8388962 |
255.2966 |
0.2 |
6.164555 |
0.8146692 |
260.1437 |
0.4 |
6.436470 |
0.7907144 |
264.9636 |
0.5 |
6.571032 |
0.7788456 |
267.3619 |
0.6 |
6.681199 |
0.7676471 |
269.7519 |
0.8 |
6.620103 |
0.7526445 |
274.5991 |
1.0 |
6.559204 |
0.7380194 |
279.5790 |
Here we have given graphs, which
shown how change the value of of
|
|
Graph 4: |
Graph 5: |
|
|
Graph 6:
4.2.3. Case-2,
Example-1:
From
the above example in 4.1.2.1, if we take the input data’s of the proposed inventory model as linear
fuzzy number, then the fuzzy numbers are
p = [13, 19] ⇒
The optimal solution of the fuzzy
model by interval-valued parametric geometric programmingis presented in Table
below:
Table
7: The Optimal Solution for Proposed Fuzzy Inventory Model
|
|
|
|
0.0 |
1.145517 |
0.7339285E-01 |
324.5610 |
0.2 |
1.214499 |
0.7258219E-01 |
321.8566 |
0.4 |
1.287479 |
0.7179232E-01 |
319.0499 |
0.5 |
1.325557 |
0.7140683E-01 |
317.6020 |
0.6 |
1.364750 |
0.7102878E-01 |
316.1204 |
0.8 |
1.446687 |
0.7029944E-01 |
313.0398 |
1.0 |
1.533798 |
0.6961592E-01 |
309.7684 |
Here we have given graphs, which
shown how change the value of
|
|
Graph 7: |
Graph 8: |
Graph 9:
4.2.4. Case-2,
Example-2:
From
the above example in 4.1.2.2, if we take the input data’s of the proposed inventory model as linear fuzzy number, then the fuzzy numbers are
p = [60, 70] ⇒
The optimal solution of the fuzzy
model by interval-valued parametric geometric programming is presented in Table
below.
Table
8: The Optimal Solution for proposed Fuzzy Inventory Model
|
|
|
|
0.0 |
7.503546 |
0.2857741 |
210.3772 |
0.2 |
7.377425 |
0.2778402 |
216.7752 |
0.4 |
7.252474 |
0.2701511 |
223.4587 |
0.5 |
7.190394 |
0.2663937 |
226.9127 |
0.6 |
7.128557 |
0.2626922 |
230.4446 |
0.8 |
7.005571 |
0.2554510 |
237.7505 |
1.0 |
6.883443 |
0.2484168 |
245.3949 |
Here we have given a rough sketch,
which shown how change the value of
|
|
Graph 10: |
Graph 11: |
Graph 12:
5. Observations
Here,
we have taken the possible values of Cost parameter, selling price and display
shelf-space in the parametric interval form as [
6. Conclusions
In
this paper, we produced a real-life E. P. Q. Inventory Model in a crisp
environment. The inventory model developed for selling price, display
shelf-space and frequency of advertisement depended on demand with time
depended holding cost and fully backlogged shortages under non-instantaneous
deterioration. Here, delay in payment is permissible. In this model, also,
production is proportional to demand and demand rate is taken as
Here,
the crisp model has produced then it changed to fuzzy model taking the linear
fuzzy number for the Cost parameters, selling price and display shelf-space and
illuminated by the nearest interval approximation method, thereafter
transformed this interval number to a parametric interval-valued functional
form and solved. This model has been
developed for the single item.
This
type of inventory model is a potential field of research. In the future, a lot
of scope for additional work based on what has been presented in this research
work. On the other side, in the fuzzy environment, the other sort of membership
functions such as piecewise linear hyperbolic Fuzzy Number, Parabolic
Fuzzy Number (pFN), Parabolic flat Fuzzy Number (PfFN), Piecewise Linear
Hyperbolic Fuzzy Number, Parabolic level Fuzzy Number (PfFN), Pentagonal Fuzzy
number and so forth can be considered to construct the membership function and
then the model can be easily solved by using Werner’s Approach, Geometric
Programming (GP) technique, Nearest Symmetric Triangular Defuzzification (NSTD)
method and so forth.
7. Acknowledgement
The authors are thankful to University of Kalyani for
providing financial assistance through DST-PURSE Programme. The authors would
like to thank the editor and anonymous reviewers for their valuable and
constructive comments and suggestions which have led to a significant
improvement in the manuscript.
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