4. Monalisha Pattnaik.pdf

*Corresponding Author: Monalisha Pattnaik, Email: monalisha_1977@yahoo.com
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Ordering and Pricing Fuzzy Optimal Replenishment Policies for Deteriorating Items with Two
Dimensional Demand under Disposal Mechanism
1
Monalisha Pattnaik*, 2
Padmabati Gahan
1, 2
*Dept. of Business Administration Sambalpur University Jyotivihar, Sambalpur India
Received on: 17/02/2017, Revised on: 04/03/2017, Accepted on: 10/03/2017
ABSTRACT
This paper tries to incorporate previous studies to develop the ordering and pricing inventory models.
That is, I want to investigate the optimal decision parameters for deteriorating items with stock and price
dependent demand under disposal mechanism. As markets have become more and more competitive,
many business practices show that the presence of a larger quantity of goods displayed may attract more
customers than that with a smaller quantity of goods essentially this study focuses pricing and ordering
strategies since the demand for the goods may be affected for a firm that sells a seasonal item over a
finite planning time. In this paper, pricing is a major strategy for a retailer to obtain its maximum profit.
This paper considers the modification of EOQ formula in the presence of imprecisely estimated system
cost i.e. holding cost and ordering cost. These imprecise parameters are presented by fuzzy numbers
defined on a bounded internal on the axis of real numbers. The main contribution to literature is the
inclusion of the fuzzy approach in continuous crisp model. Here, for the times of price changes, I give
some sufficient conditions for optimal decision rules. The analysis shows the influence of key model
parameters.
Keywords: Ordering, Pricing, Fuzzy, Disposal cost, Continuous model
INTRODUCTION
Classical inventory models are usually developed over infinite planning horizon. The assumption of an
infinite planning horizon is not realistic due to several reasons such as variations of inventory costs,
changes in product specifications and designs, technological changes etc., the business period is not
finite. Seasonal items are an important part of stocks carried in practice. The problem of managing
inventory of a seasonal product is complex for a variety reasons. The product can usually be produced by
the vendor only at finite rate. In many instances, the demand for the product is sensitive to the price
charged by the vendor to the customer.
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Asian Journal of Mathematical Sciences 2017; 1(2):52-64
To survive and thrive in the highly competitive retail world, retailers must become more attentive and
meticulous with their pricing. The financial success of companies selling retail goods depends on their
pricing strategy. In single-price strategy customers have to pay the same price for all items in entire
planning horizon. With such a policy the variety of offerings is often limited. The strength is being able
to avoid employee error and facilitate the speed of transactions. Despite the increased role of non-price
factors in the modern marketing process, price remains an important element in case of marketing
decisions. Historically, price has been the major factors affecting buyer’s choice. Price is also one of the
most flexible elements of the marketing mix as it can be increased or decreased according to need. At the
same time pricing is one of the major problems faced by many marketing executives or decision makers
as product move through their life cycle. This led many researchers to investigate inventory models with
price dependent demand. Inventory management plays an important role in business since it can help the
companies reach the goal of ensuring the prompt delivery, avoiding shortages, helping sales at

Pattnaik Monalisha et al. Ordering and Pricing Fuzzy Optimal Replenishment Policies for Deteriorating Items with Two
53
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competitive prices and so forth. Since a firm may use a pricing strategy to spur demand for its seasonal
goods, the inventory problems with price and stock dependent demand cannot be extended to a model
with a single price markdown.
Various paradigmatic changes in science and mathematics concern the concept of uncertainty. In science,
this change has been manifested by a gradual transition from the traditional view, which insists that
uncertainty is undesirable and should be avoided by all possible means. From practical experience, it has
been found that uncertainty occurs not only due to a lack of information but also as a result of ambiguity
concerning the description of the semantic meaning of declaration of statements relating to an economic
world. The fuzzy set theory was developed on the basis non-random uncertainties. For this reason, this
study considers the inventory holding cost and ordering cost as the fuzzy number, since no researchers
have discussed profit maximization inventory models for deteriorated items with price and stock sensitive
demand by introducing fuzzy disposal cost and fuzzy ordering cost over continuous finite planning
horizon. A fuzzy solution process is proposed to maximize the optimistic/pessimistic return and a fuzzy
solution based model is developed to solve the problem. The models are illustrated with some numerical
data. Some sensitivity analyses expected profit.
There is considerable literature on problem of determining the lot size of seasonal products with
deterioration under additions of finite productions and fixed demand. The production-inventory systems
of deteriorated seasonal products are most common in reality and a number of researchers have
investigated the problem of determining economic replenishment policy of such items. Bhunia and Maiti
[1] investigated a deteriorating inventory model with linear stock and time dependent demand. Giri and
Chaudhuri [2] dealt with an inventory model with poIr form stock dependent demand. Chung, Chu, Lan
[3] studied a deteriorating inventory with linear stock dependent demand. Chung [4] developed an
algorithm for an inventory system with a poIr form stock dependent demand. Teng and Chang [5]
investigated a production model with linear stock-dependent demand. Urban [6] dealt with a periodic
review inventory model under the assumption that demand is serially correlated and dependent on the
initial inventory level. Although these models offer very good insights into the literature of the lot sizing
problem and many of its aspects, they do not take into account the continuous model with deteriorated
seasonal products in finite planning horizon. Incorporating deteriorated seasonal products and stock and
price sensitive demand explicitly in the model may reveal new insights about the setting and ordering of
price relationship. Tripathy and Pattnaik [7] investigated a fuzzy entropic order quantity model for
perishable items where pre and post deterioration discounts are alloId with two component demand.
Tripathy and Pattnaik [8] extended this work with modification of the model with constant demand and
instant deterioration discount for perishable items in fuzzy decision space. Finally, most of these models
dealt with single-level lot sizing problems. Incorporating multiple price changes when the demand is
price and stock dependent for deteriorated seasonal products adds accuracy to the model and new insights
about the relationship betIen the number of price change, selling price and lot sizing.
Many inventory models have been proposed to deal with a variety of inventory problems. Comprehensive
review of inventory models can be found in Khouja, Petruzz and Dada [9, 10]. To control an inventory
system, one cannot be ignored demand monitoring since inventory is partially driven by demand, and as
suggested by Lau and Lau [11]. Tripathy and Pattnaik [12] developed a fuzzy inventory model where unit
cost of production is a function of reliability and constant demand. Researchers and practitioners are of
opinion that one of the major factors for the occurrence of variability of demand rate is due to its time
dependency. As pricing is an obvious strategy to influence demand, studies on inventory models with
price-dependent demand have received much attention. Polatoglu [13] proposed an inventory model for
developing pricing and procurement decisions simultaneously. You [14] developed an inventory model in
which the demand is price and time dependent and the number of price changes can be controlled.
Khouja [15] investigated a newsboy problem in which discount prices are decision variables and discount
strategies are used to sell excess inventory. Shinn and Hwang [16] dealt with the problem with
determining the order quantity under the condition that the demand is a convex function of price and
delay in payments is order-size dependent. You and Hsieh [17] extended an inventory model with stock
and price sensitive demand where multiple price changes are alloId.
This paper establishes and analyzes three inventory models under profit maximization which extends the
classical economic order quantity (EOQ) model. A truly efficient EOQ does more than just reduce cost. It
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also creates revenue for the retailer and the manufacturer. The evolution of the EOQ model concept tends
toward revenue and demand focused strategic formation and decision making in business operations. This
paper focuses on the profit maximizing issues in a continuous production model, based on cost reduction
mechanisms and a revenue improvement stimulus. It is noted that the literature herein rarely considers the
cases with multiple price changes when the demand is price and stock dependent. Since a firm may reset
its selling price to spur demand when consumers’ purchasing behavior are price and stock dependent, this
paper studies single replenishment inventory model to deal with this problem. The purpose of this paper
is to develop the solution procedure for determining the optimal order size and optimal selling prices for a
deteriorated seasonal item.
The remainder of the paper is organized as follows. In section 2 assumptions and notations are provided
for the development of the model. The mathematical formulation is developed in section 3. Section 4
formulates the fuzzy model to find out the optimal solution in finite horizon. In section 5, inventory
model without price change is derived and the existence of the solution is verified. The inventory model
with a single price change and the solution procedure are given 6. The inventory model with two price
changes is formulated and the solution procedure is given in section 7. In section 8, the numerical
examples are presented to illustrate the development of the three different models. The sensitivity
analysis is carried out in section 9 to observe the changes in the optimal solution. Finally section 10 deals
with the summary and the concluding remarks.
Table-1: Summary of the Related Research
Authors Demand factors Demand patterns Deterioration Planning Horizon Disposal cost Structure of the
model
Bhunia et al. (1998) Stock and time Linear Yes Finite No Crisp
Giri et al. (1998) Stock Non-linear form Yes Finite No Crisp
You et al. (2007) Stock and price Linear (sensitive) No Finite No Crisp
Tripathy et al. (2011) Stock Linear Yes Finite Yes Fuzzy
Present paper (2017) Stock and price Linear (sensitive) Yes Finite Yes Fuzzy
Assumptions and Notation
A mathematical model of the problem is developed. Suppose a firm purchases Q units of a deteriorated
seasonal item and sells them over a finite time horizon L and θ is the constant deterioration rate, where
. Demand for the item is assume to be price and stock dependent. The firm previously divides
the planning horizon L into equal time periods, each with T = L / time units. The firm set an
initial selling price at the start of period 1. At the start of subsequent periods, the firm resets the selling
price. The selling price set during period is j denoted by .
The demand rate at t of period j is assumed to follow the form of where
α is the intersection of the demand curve, the values of β and η are constants, and represents the
inventory level with time t of period j.
It is assumed that the unit time holding cost per unit is h and the unit purchasing cost is c. Changing price
may involve some costs, such as changing price lists, tags and catalogues, changing product label,
advertising price changes as Ill as communicating the logic behind the list price changes to different
firms. I assume that there is an ordering cost K associated with each price setting. The price adjustment
cost can be estimated by the sum of all the component costs. The firm aims to maximize its profit by
simultaneously determining (1) the order quantity Q and (2) the selling prices The notation
is summarized in the following.
Notations
Q: order quantity,
L: planning time interval,
n: the total number of periods (n-1 also represents the number of price changes),
T: length of a period, T = L/n,
t: period index, period j refers to the time interval [ (j-1) T, jT),
p: selling price set during period j,
λ(p,t): demand rate at time t of period j when the initial selling price is set at p,
c: unit purchasing cost,
K: pricing setting cost,
h: inventory holding cost per unit time,
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d: deterioration cost per unit time, and
θ : the deterioration rate.
Formulation
Now the model of the problem will be developed. Suppose the firm divides the sales season into n
periods. In addition, assume that the firm sets the order quantity and the selling price at Q and P = (p1,
p2,….pn), respectively. Then since the demand rate at time t of period j is
λj (t,pj) = α – βpj +ηIj(t), I have for period j.
– 0 ≤ t ≤ T (1)
Where, time points 0 and T respectively denote the starting and ending times of a period. Let qj be the
inventory level at the start of period j. Then I have Ij (0) = qj from which I obtain
(2)
Since Ij-1 (T) = Ij(0), it follows that, Ij-1 (T) =qj from which I obtain
(3)
Where To reduce the unknown term qj in (2), I will re express qj in terms of Q and P.
Since the initial inventory level is Q, I have q1 = Q. Thus, I have , Let
. Then, I have q2 = . Assume that
. Then by induction, since I can show that qj = , qj can be re-expressed as follows.
(4)
Now I will develop the profit function which is comprised of sales revenues, inventory holding cost,
purchasing cost and pricing setting cost. Substituting qj in (4) into Ij (t) in (2) I have
(5)
Sales Revenues
Let ∆qj denote the sales amount during period j. Then, I have ∆qj = qj – qj+1
=
= (6)
Let (n) be the sales revenue when the firm divides the sales season into n periods. Then, I have
(7)
Inventory Carrying Cost and Waste Disposal Cost
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Let be the carrying cost and disposal cost of period j respectively when the firm divides
the sales season into n periods.
Then,
1− −1 =2 − − −1(ℎ+ )( + )2 − 1− + − −1 − + 2(ℎ+ )
(8)
−1 =1 + (ℎ+ ) + −2 (ℎ+ ) + 21− −1
(9)
Where, and are the total carrying cost and waste disposal cost respectively when the firm
divides the sales season into n periods. Let F(n,P,Q) be the total profit when the firm divides the sales
season into n periods. Then, I have
F(n,P,Q)= (10)
Note that the inventory level at the ending time of period n is zero. Thus, I have In(T) = 0. Let Q(n) be the
solution to the equation In(T) = 0. Then, I obtain
(11)
Substituting Q= Q(n) into , and and letting F(n,P) be the result, I have
F(n,P) = R(n) – H(n) –D(n) – Q(n)c – nK (12)
Where,
(13)
+
(14)
F =F(n,pi) = F1i + F2i d+ F3iK, (n= 1, 2, 3) and (i= 1 to n) (15)
Where, F1i =
c+h
1− −12 =1 =2 − − −1 + 2+ 1− + − −1 =1 + 2− + + 2 + 21− −1
(16)
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F2i=
(17)
F3i = (- n) (18)
Fuzzy Model
I replace the holding cost and ordering cost by fuzzy numbers and respectively. By expressing and
as the normal triangular fuzzy numbers (d1, d0, d2) and (K1, K0, K2),
where,d1=d ,do=d,d2=d+ such that 0
are determined by the decision maker based on the uncertainty of the
problem. The membership function of fuzzy holding cost and fuzzy ordering cost are considered as









≤
≤
−
−
≤
≤
−
−
=
otherwise
,
0
,
,
)
( 2
0
0
2
2
0
1
1
0
1
~ d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
µ (19)









≤
≤
−
−
≤
≤
−
−
=
otherwise
,
0
,
,
)
( 2
0
0
2
2
0
1
1
0
1
~ K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
k
µ (20)
Then the centroid for and are given by
3
3
1
2
2
1
~
∆
−
∆
+
=
+
+
= d
d
d
d
M o
d
and
3
3
3
4
2
1
~
∆
−
∆
+
=
+
+
= K
K
K
K
M o
k
respectively.
For fixed values of n and p let, Z (d, . Let
2
3
1
F
K
F
F
y
d
−
−
= , 1
1
2
3
ψ
=
∆
−
∆
and 2
3
4
3
ψ
=
∆
−
∆
By extension principle the membership function of the fuzzy profit function is given by
{ }






∧







 −
−
=
∧
=
≤
≤
∈ −
)
(
)
(
)
(
~
2
3
1
~
~
~
)
(
)
,
(
)
(
)
~
,
~
(
~
2
1
1
K
F
K
F
F
y
Sup
K
d
Sup
k
d
k
k
k
k
d
y
Z
k
d
y
K
d
z
µ
µ
µ
µ
µ
(21)
( )
( )









≤
≤
−
−
+
+
≤
≤
−
−
−
−
=







 −
−
otherwise
,
0
,
,
2
3
0
2
2
3
2
2
1
1
2
1
0
2
3
1
2
1
2
3
1
~ u
K
u
d
d
F
y
K
F
d
F
F
u
K
u
d
d
F
K
F
d
F
F
y
F
K
F
F
y
d
µ (22)
3
1
2
1
1
F
d
F
F
y
u
−
−
= ,
3
0
2
1
2
F
d
F
F
y
u
−
−
= and
3
2
2
1
3
F
d
F
F
y
u
−
−
=
When and then and . It is clear that for
every . From the equations (13) and (16) the
value of '
PP may be found by solving the following equation:
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( )
1
0
2
3
1
2
1
1
0
`
1
d
d
F
K
F
d
F
F
y
K
K
K
K
−
−
−
−
=
−
−
or,
( )( ) ( )
( ) ( )
1
0
3
1
0
2
1
0
1
2
1
0
1
2
1
K
K
F
d
d
F
d
d
K
F
K
K
d
F
F
y
K
−
+
−
−
+
−
−
−
=
Therefore,
( ) ( )
)
(
' 1
1
0
3
1
0
2
3
1
2
1
1
0
`
1
y
K
K
F
d
d
F
K
F
d
F
F
y
K
K
K
K
PP µ
=
−
+
−
−
−
−
=
−
−
= , (say). (23)
When and then + and + . It is evident that for
every From the equations (13) and (16), the
value of "
PP may be found by solving the following equation:
( )
0
2
2
3
2
2
1
0
2
2
d
d
F
y
K
F
d
F
F
K
K
K
K
−
−
+
+
=
−
−
or,
( ) ( )( )
( ) ( )
0
2
3
0
2
2
0
2
2
2
1
0
2
2
2
d
d
F
d
d
F
d
d
y
d
F
F
d
d
d
F
K
−
+
−
−
−
+
−
−
=
Therefore,
( ) ( )
)
(
" 2
0
2
3
0
2
2
2
3
2
2
1
0
2
2
y
K
K
F
d
d
F
y
K
F
d
F
F
K
K
K
K
PP µ
=
−
+
−
−
+
+
=
−
−
= , (say). (24)
Thus the membership function for fuzzy total profit is given by
otherwise
;
;
;
0
)
(
)
(
)
( 2
3
2
2
1
0
3
0
2
1
0
3
0
2
1
1
3
1
2
1
2
1
)
,
~
(
~ K
F
d
F
F
y
K
F
d
F
F
K
F
d
F
F
y
K
F
d
F
F
y
y
y
K
h
z
+
+
≤
≤
+
+
+
+
≤
≤
+
+





= µ
µ
µ (25)
Now, let dy
y
P
k
d
z
∫
∞
∞
−
= )
(
)
~
,
~
(
~
1 µ and dy
y
y
R
k
d
z
∫
∞
∞
−
= )
(
)
~
,
~
(
~
1 µ hence, the centroid for fuzzy total profit is given
by (n = 1, 2, 3) and (i = 1 to n)
1
1
)
,
(
~
~
P
R
p
n
M
F i
TPn
n =
=
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
2
2
2
1
3
2
1
i
i
i
i
i
i
i
i
i
i
p
n
F
p
n
F
K
p
n
F
d
p
n
F
p
n
F
ψ
ψ +
+
+
+
=
(26)
i
i
i
i
TPn
F
K
F
d
F
p
n
M 3
2
2
1
1 )
(
)
(
)
,
(
~ ψ
ψ +
+
+
+
= (27)
where, are given by equations (9), (10) and (11).
5. Inventory Model without Price Change
In this section, I assume that the firm sets its setting price at the start of the sales season and does
not reset its selling price thereafter substituting n=1 into (15) I have
Fuzzy Total Profit
(28)
(29)
Where,
- +h(
(30)
( ) (31)
(32)
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Taking the first and second derivatives of with respect to gives
(33)
(34)
Let pj(n) be the optimal setting price for period j when the firm divides the sales season into n periods.
Since , objective function is concave in p1. Accordingly, the optimal setting price is given
by the solution in the first order condition Let (29) be equal to zero.
Then, I have (35)
Substituting p1= into (11), I have
(36)
Inventory Model with a Single Price Change
In this section, I assume that the firm sets its selling price at the start of the sales season and resets its
selling price at the time of l/2 substituting n=2 into (15) I have,
Fuzzy Total Profit
(2, p1, p2) =
(37)
(2, p1, p2) = (38)
Where,
h
( (39)
(40)
= (-2) (41)
The Hessian matrix of (2, p1, p2) is given by (42)
Note that and for x < 3. Thus, F is concave
function of selling prices when Accordingly, for x< 3, the optimal setting
prices are given by the solutions to the first order conditions
=0 (43)
Solving the above equation system gives
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(44)
(45)
Substituting and into (38), I have
(46)
Inventory Model with Two Price Changes
In this section, assuming that the firm sets its selling price at the start of the sales season and resets its
selling prices at the times of L/3 and 2/3 L, respectively. Substituting n=3 into (15) I have
Fuzzy Total Profit
+
(47)
(48)
Where,
( + ) 1 + 2+ + + + 2− 2 + 2
+h
) 1 + 2+( 2− − + ) 2 + 2 + 3− 2− + 3 + 2 +
6 ( −1) + 2−3 + − −1 2+ +1 + 2)
(49)
+ (50)
= (-3) (51)
The Hessian matrix of is given by (52)
Note that and for x < 3 and
13 −12 −2<0 for x < 2. Thus, 3, 1, 2, 3 is concave function of selling prices when x < 2.
Accordingly, for the optimal setting prices are given by the solutions to the
first order conditions. The first order conditions are
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0 (53)
=0 (54)
0 (55)
Using the above equation system yields
(56)
(57)
(58)
Substituting p1 = 1(3), P2 = 2(3) and p3 = 3(3) into (48), I obtain
(59)
Numerical Examples
Example 1
Suppose L=120, h=0.005, d=2.0, α = 50, β=1.5, η = 0.01, c = 20 and K = 500, θ = 0.002, ψ1=1.0 and
ψ2=10. It is noted that . Then, from (35) and (36) I have
Table-2 Optimal Values for n=1 (T=L)
Fuzzy
Optimal Values 26.7673 2581.894 10867.98
Example 2
Suppose the parameters are the same as Example-1. It is noted Thus
the optimal setting prices are given by the solutions to the first order conditions. From (44) – (46) it is
obtained
Table-3 Optimal Values for n = 2 (T=L/2)
Fuzzy
Optimal Values 32.13626 19.99695 3544.06 10127.77
Example 3
Suppose the parameters are the same as Example-1. It is noted . Thus
the optimal selling prices are given by the solutions to the first order conditions. From (56) – (59) it is
obtained
Table-4 Optimal Values for n = 3 (T=L/3)
Fuzzy
Optimal Values 36.89843 26.88428 16.97376 3818.375 25823.32
Fuzzy Sensitivity Analysis
I refer to the data set used in the above example as the basis data set, W, where W = {L = 120, h = 0.005,
d=2.0, α = 50, β = 1.5, η = 0.01, c = 20, K = 500, = 510, θ = 0.002}. I investigate the changes
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in the optimal decision values of 1(3), 2(3), 3(3), (3) and (3, 1(3), 2(3), 3(3)) when only one
parameter in the set W changes and other remain unchanged the computational results are described in
Table 5.
Table 5 Sensitivity Analysis
Parameter 1(3) 2(3) 3(3) (3) (3)
L
130 40.2697 26.89267 13.63685 5083.034 34312.33
140 45.66169 26.89966 8.277724 7105.743 47569.15
150 55.67863 26.90528 -1.70785 10863.09 71659.29
160 80.76482 26.90955 -26.7641 20271.68 130901
170 258.4707 26.91251 -204.441 86912.9 545531.4
0.012 36.85606 26.90407 17.06511 3800.802 25885.38
0.013 36.81369 26.92385 17.15647 3783.229 25948.57
0.014 36.77132 26.94363 17.24782 3765.656 26012.88
0.015 36.72895 26.96342 17.33917 3748.083 26078.31
0.016 36.68658 26.9832 17.43053 3730.511 26144.86
α
55 41.23954 28.55095 15.96597 4821.295 40534.6
60 45.58066 30.21762 14.95819 5824.215 58588.96
65 49.92178 31.88428 13.9504 6827.134 79986.38
70 54.2629 33.55095 12.94261 7830.054 104726.9
75 58.60402 35.21762 11.93483 8832.973 132810.4
η
0.007 29.76844 26.93034 24.15344 1946.248 13723.13
0.008 30.57609 26.921 23.33573 2160.101 15121.9
0.009 31.58155 26.91173 22.32026 2425.323 16847.67
0.01 32.86579 26.90251 21.02607 2762.938 19033.97
0.011 34.56097 26.89336 19.32101 3207.264 21898.7
c
21 36.59609 27.38428 18.27609 3517.5 22155.38
22 36.29375 27.88428 19.57843 3216.624 18788.32
23 35.99142 28.38428 20.88076 2915.748 15722.13
24 35.68908 28.88428 22.1831 2614.872 12956.82
25 35.38675 29.38428 23.48544 2313.996 10492.39
θ
0.013 40.32328 26.87529 13.53918 4711.863 31541.76
0.014 45.81553 26.86638 8.037297 6142.216 40669.87
0.015 56.03854 26.85755 -2.19525 8801.069 57599.06
0.016 81.6737 26.84883 -27.8399 15462.22 99940.95
0.017 263.394 26.8402 -209.57 62656.35 399638.9
In Example 3, the firm has two chances to adjust its selling prices. Table 5 shows that the initial selling
price 1(3) and secondary selling price 2(3) increase in to the length of L, and the final selling price
3(3) decreases in L. This phenomenon can be explained as follows. First, consider the situation in which
the firm has a longer selling season as situation-1 and the situation in which the firm has a shorter selling
season as situation-2. Note that the firm would expect to obtain higher unit profit. Compared with
situation-2, situation-1 gives the firm longer time to achieve this goal. Thus it can be conjectured that the
firm in situation-1 may set 1(3) and 2(3) to achieve higher unit profit. It can also be conjectured that the
remaining inventory in situation-1 may be larger than that of situation-2 when 3(3) is to be set. To, sell
out its inventory, the firm in situation-A would set a loIr 3(3) to sell its items. According to this reason,
1(3) and 2(3) increase in the length of L and 3(3) decreases in L. Finally, it is observed from the Table
5 that only 2(3) is insensitive to changes in parameter L whereas other optimum decision variables are
highly sensitive to changes in the parameter L.
It shows that 1(3) decreases in the value of , and 2(3) and 3(3) increase in the value of . this
phenomenon could be explained as follows. First, I consider the situation in which the firm has a higher
disposal cost as situation-3 and the situation in which the firm has a loIr as situation-4. Note that the
firm would shorten its and it has a tight relationship with inventory level, inventory holding time and .
Compared with situation-4, situation-3 gives the firm more incentives to reduce its inventory when the
time to sell its items is still long. It can be conjectured that the firm in situation-3 may set a loIr initial
selling price to reduce its inventory. It can also be conjectured that the remaining inventory in situation-3
may be less than that in situation-4, when 2(3) is to be set. To obtain a higher unit profit, the firm then
would set higher 2(3) and 3(3) to sell its items. According to this reason, 1(3) decreases in , 2(3) and
3(3) increase in . Hence it is seen all the decision variables are very low sensitive or insensitive to
changes in the parameter .
Table 5 shows that 1(3), 2(3) and 3(3) decrease in the value of α. This phenomenon could be explained
as follows. First, increasing the value of α moves the demand curve up. Thus, compared with an
inventory system with a smaller value of α, the firm in situation-3 with a higher value of α may set higher
selling prices to improve its unit profit. According to this reason, 2(3) and 3(3) decrease in α.
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The optimum ordering quantity (3) and the total profit (3, , , ) respectively
are highly sensitive whereas other decision variables are moderately sensitive to changes in the
parameter α.
It shows that 1(3) increases in the value of η, and 2(3) and 3(3) decrease in the value of η. This
phenomenon could be explained as follows. First I consider the situation in which the firm has a higher
value of η as situation-5 and the situation in which the firm has a loIr value of η as situation-6. Note that
the term of η I (t) has positive impact on demand. Initially the inventory level for an inventory system is
high. Under the same selling price, the demand rate in situation-5 is higher than that in situation-6. Thus,
it can be conjectured that the firm in situation-5 may have more incentives to set a higher selling price to
obtain higher unit profit. Once the secondary selling price is set, the firm in situation-5 may have more
stock on hand. To reduce its inventory, it can be conjectured that the firm would set a loIr secondary and
3(3) final selling prices to reduce its inventory in situation-5. According to this reason, 1(3) increases in
, and 2(3) and 3(3) decrease in h. It shows that 2(3) is insensitive to changes in parameter η whereas
other decision variables are highly sensitive to changes in parameter η.
From Table 5, I see that 1(3), 2(3) and 3(3) increase in the value of c. This phenomenon could be
explained as follows. First, note that increasing the value of c reduces the unit profit. Thus, compared
with an inventory system with a smaller value of c, the firm with higher value of c may set higher selling
prices to cover its unit cost and improve its unit profit. According to this reason, 1(3), 2(3) and
increase in α. Hence all the decision variables are moderately sensitive to changes in parameter c whereas
the optimal total profit (3) is highly sensitive to changes in parameter c.
It shows that 1(3) and 2(3) increase in θ, and 3(3) decrease in θ. This phenomenon could be explained
as follows. First, I consider the situation in which the firm has a longer selling season as situation-1 and
the situation in which the firm has a shorter selling season as situation-2. Note that the firm would expect
to obtain higher unit profit. Compared with situation-2, situation-1 gives the firm longer time to achieve
this goal. Thus it can be conjectured that the firm in situation-1 may set higher initial and secondary
selling prices to achieve higher unit profit. It can also be conjectured that the remaining inventory in
situation-1 may be larger than that of situation-2 when the 3(3) is to be set. To, sell out its inventory, the
firm in situation-1 would set a loIr 3(3) to sell its items. According to this reason, 1(3) and 2(3)
increase in θ and 3(3) decreases in θ. Finally, it is observed from the Table 10 that only 2(3) is
insensitive to changes in parameter θ whereas other optimum decision variables are highly sensitive to
changes in the parameter θ.
It is observed that the ordering quantity and the profit increase with L, α, η and θ decreases in and c. it
could be explained as follows. As L and θ increase, the firm has more time to sell its items. Thus the firm
orders more and gets more profit. Since the parameters of α and η have positive effect on demand, the
ordering quantity and the profit increase as α and η increase. Since the parameters of and c have
negative effect on demand, the ordering quantity and the profit decrease when and c increase. The
characteristics of the sensitivity analysis are summarized as follows. (1) 1(3) increases with L, α, η, c
and θ while it decreases with ; (2) 2(3) increases with L, , α and c while it decreases with η; (3)
increases with , α and c while it decreases with L, η and θ; (4) Q (3) increases with L, α ,η and θ
while it decreases with h and c; and (5) (3, , , ) increases L, α , η and θ while it
decreases with h and c.
CONCLUSION
This research investigates pricing and ordering problem for deteriorating items allowing multiple price
changes and enabling control of the number of price changes into the analysis but they are more realistic
and therefore can justifiably be used in actual situations. Numerous inventory models have addressed this
problem. HoIver, these models have rarely considered a situation in which the sales price can be adjusted
during the selling period and the number of price changes can be controlled. Taking this situation into
account, this paper developed a continuous time fuzzy model with disposal cost and ordering cost for
finding optimal ordering quantity and pricing setting/changing strategy. The necessary and sufficient
conditions of the existence and uniqueness of the optimal solutions under various cases are presented in
the paper. Finally, numerical examples have been given to illustrate the theoretical results, with
consequent managerial implications. These models in this study are general frameworks which consider
price setting and changing strategies simultaneously.
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I believe that our work will provide a basic foundation for the further study of this kind of important
inventory models with fuzzy space. Hence the utilization of fuzzy decision space on the costs makes the
scope of the application broader.
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