Abstract
In this paper, we have analysed a deterministic inventory model for
deteriorating items with timedependent quadratic demand and holding cost is timedependent.
An Exponential distribution is used to represent the distribution of
time to deterioration. In the model considered here, shortages
are allowed and partially backlogged. The backlogging rate is assumed to be dependent
on the length of waiting for the next replenishment. The longer the waiting time
is, the smaller the backlogging rate would be. The model is solved analytically
to obtain the optimal solution of the problem. The derived model is illustrated
by a numerical example and its sensitivity analysis is carried out.
Keywords
Deteriorating items,Exponential distribution,Inventory,
Partial backlogging, Quadratic demand, Shortages,Timevarying holding cost.
1. Introduction
Inventory system is one of the main streams of the Operation Research which is essential
in business enterprises and industries.Inventory may be considered as an accumulation
of a product that would be used to satisfy future demands for that product. It needs
scientific way of exercising inventory model.
An optimal replenishment policy is dependent on ordering cost, inventory carrying
cost and shortage cost. An important problem confronting a supply manager in any
modern organization is the control and maintenance of inventories of deteriorating
items. Fortunately, the rate of deterioration is too small for items like steel,
toys, glassware, hardware, etc. There is little requirement for considering deterioration
in the determination of economic lot size.
Inventory of deteriorating items first studied by Whitin (1957), he considered the
deterioration of fashion goods at the end of prescribed storage period. Ghare and
Schrader (1963) extended the classical EOQ formula to include exponential decay,
wherein a constant fraction of on hand inventory is assumed to be lost due to deterioration.Covert
and Philip (1973) and Shah and Jaiswal (1977) carried out an extension to the above
model by considering deterioration of Weibull and general distributions respectively.
Dave and Patel (1981)first developedan inventory model for deteriorating items with
time proportional demand, instantaneous replenishment and no shortagesallowed. Many
researchers such as Park (1982)and Hollier and Mak (1983) also considered constant
backlogging rates in their inventory models. Nahmias (1978) gave a review on perishable
inventory theory. Rafaat (1991) described survey of literature on continuously deteriorating
inventory model. He focused to present an uptodate and complete review of the
literature for the continuously deteriorating mathematical inventory models.
All researchers assume that during shortage period all demand either backlogged
or lost. In reality, it is observed that some customers are willing to wait for
the next replenishment. Abad (1996) considered this phenomenon in his model, optimal
pricing and lot sizing under conditions of perishable and partial backordering.
He assume that the backlogging rate depends upon the waiting time for the next replenishment.
But he does not include the stock out cost (back order cost and lost sale cost).
Goyal and Giri (2001) gave recent trends of modeling in deteriorating inventory.
Ouyang, Wu and Cheng (2005) developed an inventory model for deteriorating items
with exponential declining demand and partial backlogging. Dye and Ouyang (2007)
found an optimal selling price and lot size with a varying rate of deterioration
and exponential partial backlogging. They assume that a fraction of customers who
backlog their orders increases exponentially as the waiting time for the next replenishment
decreases. Singh and Singh (2007) presented an EOQ inventory model with Weibull
distribution deterioration, Ramp type demand and Partial Backlogging. NitaShah and
Kunal Shukla (2009) developed a deteriorating inventory model for waiting time partial
backlogging when demand is constant and deterioration rate is constant. Singh,T.J.,
Singh, S.R. and Dutt, R. (2009) extended an EOQ model for perishable items with
power demand and partial backlogging.Skouri, Konstantaras, Papachristos and Ganas
(2009) developed an Inventory models with ramp type demand rate, partial backlogging
and Weibell's deterioration rate.
An exponentially timevarying demand also seems to be unrealistic because an exponential
rate of change is very high and it is doubtful whether the market demand of any
product may undergo such a high rate of change as exponential.
In reality, the demand and holding cost for physical goods may be time dependent.
Time also plays and important role in the inventory system. So, in this paper we
consider that demand and holding cost are time dependent.
Recently, Mishra and Singh (2011) developed a deteriorating inventory model with
partial backlogging when demand and deterioration rate is constant. Vinodkumar Mishra
(2013) developed an inventory model of instantaneous deteriorating items with controllable
deterioration rate for time dependent demand and holding cost.
J. Jagadeeswari and P. K. Chenniappan (2014) developed an order level inventory
model for deteriorating items with time – quadratic demand and partial backlogging.
Sarala Pareek and Garima Sharma (2014) developed an inventory model with Weibull
distribution deteriorating item with exponential declining demand and partial backlogging.
R. Amutha and Dr. E. Chandrasekaran developed an inventory model for deteriorating
items with time – varying demand and partial backlogging.Kirtan Parmar and U. B.
Gothi(2014) developed a deterministic inventory model for
deteriorating items where time to deterioration has Exponential distribution and
with timedependent quadratic demand. In this model, shortages are not allowed and
holding cost is timedependent. Also, U. B. Gothiand Kirtan Parmar(2015)have extended above deterministic inventory model by taking
two parameter Weibull distribution to represent the distribution of
time to deterioration and shortages are allowed and partially
backlogged.Kirtan Parmar and U. B. Gothi (2015) developed an economic
production model for deteriorating items using three parameter Weibull distribution
with constant production rate and time varying holding cost.
In this paper, we have analysed an inventory system order level lot size model for
deteriorating items under quadratic demand and time dependent IHC.
2. Notations
The mathematical model is developed using the following notations:
01. Q(t) : The instantaneous state of the inventory level at any time t. (0 ≤
t ≤ T)
02. R(t) : Quadratic demand rate.
03. A : Ordering cost per order.
04. C_{h} : Inventory holding cost per unit per unit time.
05. C_{d} : Deterioration cost per unit per unit time.
06. Cs : Shortage cost due to lost sales per unit.
07. Q : Order quantity in one cycle.
08. p_{c} : Purchase cost per unit.
09. l : Opportunity cost due to lost sales per unit.
10. t_{1} : The time at which the inventory level reaches zero (decision
variable)
11. T : Length of cycle time (decision variable).
12. IM : The maximum inventory level during [0, T].
13. IB : The maximum inventory level during shortage
period.
14. TC(t_{1},T) : Total cost per unit time.
3. Assumptions
The model is derived under the following assumptions.
1. The inventory system deals with single item.
2. The annual demand rate is a function of time and it is R(t) = a+bt+ct^{2}
(a, b, c > 0)
3. Holding cost is linear function of time and it is C_{h} = h + rt (h,
r > 0)
4. The lead time is zero.
5. Time horizon is finite.
6. No repair or replacement of the deteriorated items takes place during a given
cycle.
7. Total inventory cost is a real, continuous function which is convex to the origin.
8. Shortages are allowed and partially backlogged.
During stock out period, the backlogging rate is variable and is dependent on the
length of the waiting time for the next replenishment. The backlogging rate is assumed
to be
where
the backlogging parameter δ (0< δ<1) is a positive constant and
(T – t) is waiting time (t_{1} ≤ t ≤ T).
4. Mathematical Model And Analysis
Here, the replenishment policy of a deteriorating item with partial backlogging
is considered. The objective of the inventory problem is to determine the optimal
order quantity and the length of ordering cycle so as to keep the total relevant
cost as low as possible. The behavior of inventory system at any time is shown in
Figure 1.
Replenishment is made at time t = 0 and the inventory level is at its maximum level
S. During the period [0, μ] the inventory level is decreasing and at time t_{1}
the inventory reaches zero level, where theshortages starts and in the period [t_{1},
T] some demands are backlogged.
The pictorial representation is shown in the Figure 1.
As described above, the inventory level decreases owing to demand rate as well as
deterioration during inventory interval [0, t_{1}]. Hence, the differential
equation representing the inventory status is given by
(1)
(2)
During the shortage interval [t_{1},T], the demand at time t is partly backlogged
at the fraction
.
Thus, the differential equation governing the amount of demand backlogged is as
below.
(3)
The boundary conditions are Q(0) = S, Q(t_{1}) = 0 and Q(T) = 0. (4)
Using the boundary condition Q(0) = S the solution of equation (1) is
(5)
Similarly, the solution of equation (2) is given by
(neglecting higher powers of θ)
(where
which is obtained using Q(t_{1}) = 0)
(μ ≤ t ≤ t_{1}) (6)
In equations (5) and (6) values of Q(t) should coincide at t = μ, which implies
that
(7)
Solution of equation (3) is given by
(8)
(where k_{1} is the constant of integration and
)
With boundary condition Q(t_{1}) = 0, we get
(9)
Therefore, from (8) and (9)
(t_{1}
≤ t ≤ T) (10)
The total cost comprises of following costs
1) The ordering cost
OC = A (11)
2) The deterioration cost during the period [μ, t_{1}]
(12)
3) The inventory holding cost during the period [0, t_{1}]
(13)
4) The shortage cost per cycle
5) Lost sales cost per cycle
The maximum backordered inventory is obtained at t = T and it is denoted by IB.
Then from equation (10),
IB = – Q(T)
(16)
Thus, the order size during total interval [0, T] is given by
6) Purchase cost per cycle
(17)
Hence the total cost per unit time is given by
(18)
Our objective is to determine optimum value of t_{1} and T so that TC(t_{1},T)
is minimum. The values of t_{1} and T, for which the total cost TC(t_{1},T)
is minimum, is the solution of equationssatisfying
the condition
The optimal solution of the equations (18) can be obtained by using appropriate
software. This has been illustrated by the following numerical example.
5. Numerical Example
We consider the following parametric values for A = 300, a = 10, b = 8, c = 5, h
= 1, r = 0.5,
μ = 1, θ = 0.02, δ = 0.03, C_{d} = 5, p_{c} = 15,
ℓ = 10,
Cs = 2.
We obtain the optimal value of t_{1} = 0.9483421102 units, T = 1.577867692
units and optimal total cost (TC) = 558.4065267.
6. Sensitivity Analysis
Sensitivity analysis depicts the extent to which the optimal solution of the model
is affected by the changes or errors in its input parameter values. In this section,
we study the sensitivity of the total cost per time unit TC(t_{1},T) with
respect to the changes in the values of the parameters A,
a, b, c, h, r, δ, θ, μ, Cd, Cs,ℓ,
pc.
The sensitivity analysis is performed by considering 10% and 20% increase or decrease
in each one of the above parameters keeping all other parameters the same. The results
are presented in Table – 1.
Table – 1 :Partial Sensitivity Analysis
Parameter

% change

t_{1}

T

TC(t_{1}, T)

% changes in TC(t_{1}, T)

A

– 20

0.8993971118

1.477499780

519.1382696

–7.0322

– 10

0.9246145913

1.529103772

539.0960141

–3.4581

+ 10

0.9707711691

1.624150091

577.1373347

3.3543

+ 20

0.9920569350

1.668242082

595.3662457

6.6188

Parameter

% change

t_{1}

T

TC(t_{1}, T)

% changes in TC(t_{1},
T)

a

– 20

0.9419996943

1.564815698

520.9298347

–6.7114

– 10

0.9451905158

1.571380204

539.6683487

–3.3557

+ 10

0.9514556274

1.584279939

577.1266315

3.3524

+ 20

0.9545315317

1.590618679

595.8378885

6.7032

b

– 20

0.9710400992

1.624703595

535.9056406

–4.0295

– 10

0.9594533014

1.600770497

547.2206714

–2.0032

+ 10

0.9376832213

1.555936104

569.4499901

1.9777

+ 20

0.9274506697

1.534920949

580.3918423

3.9372

c

– 20

0.9820435400

1.647479482

543.8875172

–2.6001

– 10

0.9643731432

1.610928095

551.2835994

–1.2756

+ 10

0.9336946205

1.547742541

565.2626456

1.2278

+ 20

0.9202298026

1.520117379

571.9070739

2.4177

δ

– 20

0.9554155989

1.577174227

558.5077790

0.0181

– 10

0.9512627264

1.577519711

558.4531831

0.0084

+ 10

0.9453947850

1.578217721

558.3463806

–0.0108

+ 20

0.9424215236

1.578569962

558.2897373

–0.0209

θ

– 20

0.9462115569

1.577699809

558.41263880

0.0011

– 10

0.9472662105

1.577783772

558.40855210

0.0004

+ 10

0.9494386876

1.577951514

558.39323820

–0.0024

+ 20

0.9505554036

1.578035177

558.38756340

–0.0034

μ

– 20

0.9070712944

1.541352795

544.8061644

–2.4356

– 10

0.9274092940

1.559030275

551.3526329

–1.2632

+ 10

0.9698753887

1.597883308

565.9827906

1.3568

+ 20

0.9920114179

1.619088991

574.1176913

2.8136

C_{d}

– 20

0.9356508387

1.552691306

548.3957130

–1.7927

– 10

0.9420604227

1.565368728

553.4206601

–0.8929

+ 10

0.9545002136

1.590194233

563.3438792

0.8842

+ 20

0.9605388222

1.602354067

568.2564563

1.7639

Cs

– 10

0.9145209509

1.581850958

557.7246487

–0.1221

+ 10

0.9783947776

1.574229668

559.0180047

0.1095

+ 20

1.0052834910

1.570894744

559.5802258

0.2102

Parameter

% change

t_{1}

T

TC(t_{1}, T)

% changes in TC(t_{1},
T)

ℓ

– 20

0.9386232686

1.579024205

558.2065776

–0.0358

– 10

0.9435260803

1.578441995

558.3086133

–0.0175

+ 10

0.9530733266

1.577301089

558.4983190

0.0164

+ 20

0.9577237881

1.576742028

558.5961112

0.0340

p_{c}

– 20

1.0234895300

1.699944268

497.3200246

–10.9394

– 10

0.9835438312

1.635033903

527.8632756

–5.4697

+ 10

0.9170487491

1.526998685

588.5938469

5.4060

+ 20

0.8890286034

1.481332007

618.1363389

10.6965

h

– 20

1.0041480650

1.580697438

557.3441158

–0.1903

– 10

0.9754911938

1.579225051

557.8895890

–0.0926

+ 10

0.9225918384

1.576613693

558.8793596

0.0847

+ 20

0.8981410102

1.575452603

559.3223506

0.1640

r

– 20

0.9612957349

1.578684770

558.2337262

–0.0309

– 10

0.9547326077

1.578268909

558.3193534

–0.0156

+ 10

0.9421160079

1.577480296

558.4817594

0.0135

7. Graphical Representation
Figure – 2
Figure –3
8. Conclusions
Ø It is observed from Figure
– 2 that total cost per unit time (TC) is highly sensitive to changes in
the value of p_{c}, moderately sensitive to changes in the values of A,
a and less sensitive to changes in the values of b, c, μ, C_{d}.
Ø It is observed from Figure
– 3 that total cost per unit time (TC) is highly sensitive to changes in
the values of h and Cs, moderately sensitive to changes in the values of ℓ,
r, δ and less sensitive to change in the values of θ.
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