Inventory Model for Perishable Items with Deterioration and Time Dependent Demand Rate

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 09 Issue: 08 | Aug 2022 www.irjet.net p-ISSN: 2395-0072
© 2022, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page 1058
Inventory Model for Perishable Items with Deterioration and Time
Dependent Demand Rate
Shakib Ali1, R.B. Singh2
1PG student, Department of Mathematics, Monad University, Hapur
2Professor, Department of Mathematics, Monad University, Hapur
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - In this paper, an inventory management of the
perishable items is needed in order to avoid therelevantlosses
due to their deterioration. This research work develops an
inventory model for perishable items, constrained by both
physical and freshness condition degradations. By working
with perishable items that eventually deteriorates, this
inventory model also takes into consideration the expiration
date, a salvage value, and the costofdeterioration. Inaddition,
the holding cost is modelled as a quadratic function of time.
The proposed inventory model jointly determines the optimal
price, the replenishment cycle time, and the order quantity,
which together result in maximum total profitperunitoftime.
The inventory model has a wide application since it can be
implemented in several fields such as food goods (milk,
vegetables, and meat), organisms, and ornamental flowers,
among others. Some numerical examples are presented to
illustrate the use of the inventory model. The resultsshowthat
increasing the value of the shelf-life results in an increment in
price, inventory cycle time, quantity ordered, and profits that
are generated for all price demand functions. Finally, a
sensitivity analysis is performed, and several managerial
insights are provided.
Key Words: Inventory, Perishable Items, Deterioration,
Salvage value, Optimal price, Replenishment cycle time.
1.INTRODUCTION
Deterioration of physical goods is one of the important
factors in any inventory and production system.Ifthe rateof
deterioration is very low its effect can be ignored. But in
many practical situations deterioration plays an important
role. It is important to control and maintain the inventories
of decaying items for the model corporation. Most of the
physical goods undergo decay or deterioration over time.
Commodities such as fruits, vegetables-andfoodstuffssuffer
from depletion by direct spoilage while kept in store. Highly
volatile liquids such as alcohol, gasolines etc. undergo
physical depletion over time through the process of
evaporation. Electronic goods, photographic film, grain,
chemicals, pharmaceuticals etc. deteriorate through a
gradual loss of potential or utility with the passage of time.
Thus deterioration of physical goods in stock isveryrealistic
feature. Deterioration rate of any item is either constant or
time dependent. When deteriorationistimedependent,time
is accompanied by proportional loss in the value of the
product. Realization of this factor motivated modelers to
consider the deterioration factor as one of the modeling
aspects.
Mashud et al. [1] determined the optimal replenishment
policy of deteriorating goods for the classical newsboy
inventory problem by considering multiple just-in-time
deliveries. Additionally, Mashud et al. [2] derived an
inventory model for deteriorating products that calculates
the optimal vales for replenishment time, price, and green
investment cost.
Given that the inherent perishability can occur immediately,
Pal et al. [3] addressed a production-inventory model for
deteriorating productswhen theproductioncostdependson
both production order quantity and production rate.Bhunia
A.K and M. Maiti, [4] were the first proponent for developing
a deterministic inventory model fordeterioratingitemswith
finite rate of replenishment dependent on inventory level.
Cheng TCE [5] discussed an economic order quantity model
with demand-dependent unit production costandimperfect
production processes. In most of the inventory models,
authors have taken constantrateofdeterioration.Inpractice
it can be observed that constant rate of deterioration occurs
rarely. Most of the items deteriorate as fast as the time
passes. It can be noticed that deteriorationdoesnotdepends
upon time only: it can be affected due to weather conditions,
humidity, and storage conditions etc. Therefore it is much
more realistic to consider deterioration rate as two or three
parameter Weibull distribution function. Many researchers
considered the constant demand rate in their inventory
models, but the assumption of constant demand is not
always applicable in real situations.
Tirkolaee et al. [8] noted that the inherent perishability
widely occurs in food goods (e.g., milk, vegetables, and
meat), organisms, and ornamental flowers. These authors
also stated that the time window between preparation and
sales of perishable items is very significant for producers
and purchasers. Giri et al. [9] noted such demand patternfor
fashionable products which initially increases exponentially
with time for a period of time after that it becomes steady
rather thanincreasing exponentially.Theramp-typedemand
demonstrates a time period classified demand pattern. In
different time periods the demand is either constant or its
rate of change is different.

According to Yavari et al. [13], one of the challenges when
managing inventories is the inherent perishability of many
items, which means their freshness and quality decrease
over time, and these cannot be sold after their expiration
date. Many researchers have modified inventory policies by
considering the "time proportional partial backloggingrate".
The present section deals with a mathematical model of tie
economic order quantity for finite time horizon. This
inventory model has been developed for a single
deteriorating item byconsideringtheconstant-deterioration
rate. In this model, a new demand pattern has been
introduced in which thedemandfollowsquadratic patternin
the initial period of time and becomes linear later on. It is
believed that such type of demand is quiterealistic fornewly
launched product in the market. Shortages are allowed and
the backlogging rate is dependentonthedurationon waiting
time and taken as exponential decreasing function of time.
The effect due to changeofdeteriorationparameterhasbeen
considered for different parameters numerically.
2. ASSUMPTIONS AND NOTATIONS
To develop inventory models for perishable items with
variable demand and partial back logging. The following
notations and assumption are used.
(i) I (t) be the inventory level at time t, t 0

(ii) 1
t is the time at which shortage starts and T is the
length of replenishment cycle 1
0 t T
  .
(iii) Replenishment rate is infinite and lead time is zero.
(iv) There is no repair OR replenishment of deteriorated
units during the period.
(v) A single item is considered over the prescribed period
T units of time.
(vi) S be the initial inventory level after fulfilling
backorders.
(vii) In this model  be the constant rate of defective units
out of on hand inventory at any time, 0
  .
(viii) Demand rate D(t) is assumed to be a function of time
such that
     
 
D t a bt c t t H t t
      where
 
H t  is the Heaviside’s functions defined as
follows.
 
1 if t
H t
0 if t
 
 
   
 
 
and ‘a’ is the initial rate of demand, b is the rate with
which the demand rate increase. The rate of change in
demand rate itself increase at a rate ‘c’. a, b and c are
positive constant.
(ix) Unsatisfied demand is back logged at a rate exp
 
x
 , where x is the time up to next replenishment.
The backlogging parameters  is a positive constant.
(x) 1 2 3
C ,C ,C and 4
C are the holding costper unittime,
unit purchase cost per unit, shortage cost per unit per
unit time and the unit cost of lost sales respectively. C’
is the inventory ordering cost per order.
3. FORMULATION AND SOLUTION OF THE MODEL
Figure 1
The depletion of inventory duringtheinterval  
1
0,t isdue
to joint effect of demand and deteriorating of items and the
demand is partially backlogged in the interval  
1
t ,T . The
depletion of inventory is given in the (figure 1).
The governing deferential equations of the proposed
inventory system in the interval (0, T) are
     
2
I t I t a bt ct
       , 0 t
   … (2.1)
     
I t I t a kt
      , 1
t t
   … (2.2)
    t
I t a kt e
    , 1
t t T
  … (2.3)
where k b c
  
with the conditions
   
1
I 0 S and I t 0
  … (2.4)
Solution of the equation (2.1)

 
   
2
dI t
I t a bt ct
dt
     
dt t
I F e e
 

   
Now
 
 
t t t
t t t
0 0 0
d e I t dt a e dt t e dt
  
   
  
 
t
2 t t
0
C t e dte I t S
 
 

t t t
t t t
2
0 0 0
a b b
e t e e
  
     
    
     
  
t t t
2 t t t
2
0 0 0
C C2 b
t e t e t e
  
     
    
     
 

t t t
t 2 t t
2 2
0 0 0
b C C2
e t e t e
  
     
   
     

 
t
t
3 0
2C
e
 
  

 
t t t
a b
e I t S e 1 te
  
   
    
   
 
t
2
b
e 1

 
 
 

2 t
C
t e
 
 
 

t
2
2C
te
 
 

t
3
2C
e 1

 
 
 

2
t
2 2 3
a bt b ct 2ct 2c
e  
      
 
  
  
 
2 3
a b 2c
  
  
t
2 2 2 2
3
e
a b t b c t 2c t 2c

 
           
 

2
3
1
a b 2c
 
    
 

t
2 2 2 2
3
e
a b t b c t 2c t 2c

 
          
 

2
3
1
a b 2c
 
    
 

   
2 t
3
1
I t S a b 2c e
 
    
 

 
 
 2
3
1
a b t 1
     

 
2 2
c t 2 t 2
     , 0 t
   …(2.5)
Now the solution of the equation (2.2)
 
   
dI t
I t a kt
dt
    
 
   
1 1
t t
t t
t t
d e I t dt a kt e dt
 
   
 
 
1 1
t t
t t t
t t
e I t 0 ae dt k t e dt
  
    
 
1 1 1
t t t
t t t
2
t t t
a k k
e t e e
  
     
    
     
  
1
t
t 1
2 2
a kt k a t k k
e e
    
      
   
   
 
   
   
t
t
2
e
e I t a k t k


     

 
1
t
1
2
e
a k t k

    

   
   
1
t t
1
2
1
I t a k t 1 e
 
   

 
 
2
1
a k t 1
    

,  
1
t t
   …(2.6)
By the equations (2.5) and (2.6) we getinitial inventorylevel
after fulfilling backorders (S)

 
   
1
t t
1
2
1
a k t 1 e
 
   

 
 
2
1
a k t 1
    

 
3
2 t
1
S a b 2c e
 
 
     
 

 
 
 
 2
3
1
a b t 1
     

 
2 2
c t 2 t 2
    
Multiplying on both sides by
t
e
 
2
3
1
S a b 2c
     

   
 
t
2 2 2
3
e
a b t 1 c t 2 t 2

          

1
t
2
e


 
 
1
a k t 1
    
  t
2
1
a k t 1 e
    

 
2
3
1
a b 2c
     

 
  1
t
1
2
1
a k t 1 e
    

 

t
2
3
e
a b t 1

     

  
2 2 2 2
c t 2 t 2 a k t k
          
 
2
3
1
a b 2c
     

 
  1
t
1
2
1
a k t 1 e
    

t
2 2 2
3
e
b t b c t


     


2 2
2c t 2c tb tc b c 
        
Putting t  
 
2
3
1
S a b 2c
     

 
  1
t
1
2
1
a k t 1 e
    


t
2 2 2
3
e
b b c t

      

2 2 2
2c 2c b c b c 
         
 
2
3
1
S a b 2c
     

 
  1
t
1
2
1
a k t 1 e
    

 
3
e
2c c

  

 
2
3
1
S a b 2c
     

 
  1
t
1
2
1
a k t 1 e
    

 
3
c
2 e
   

… (2.7)
Using (2.7) in (2.5), we get
   
2 t
3
1
I t a b 2c e
     

 
   
1
t 1
1
2
1
a k t 1 e
 
    

   
t
3
c
2 e
 
  

 
2 t
3
1
a b 2c e
    

   
 
2 2 2
3
1
a b t 1 c t 2 t 2
          

   
   
1
t t
1
2
1
I t a k t 1 e
 
   

   
 

t 2
3 3
c 1
2 e a b t 1
 
        
 
 
2 2
c t 2 t 2
     , 
0 t
   … (2.8)
Now the solution of the equation (2.4)

 
  t
dI t
a kt e
dt

  
    t
dI t a kt e dt

  
 
 
1 1 1
t t t
t t
t t t
d I t dt a e dt k t e dt
 
   
  
 
1
t
t
t
a
I t 0 e
 
   
 1 1
t t
t t
2
t t
k k
t e e
 
   
  
   
 
1 1
t t
t t
1
a k
e e t e t e
 
 
   
    
   
 
1
t
t
2
k
e e

 
 
 

t
2
a tk k
e  
  
 
  
 
1
t 1
2
a t k k
e  
   
 
  
 
   
  t
2
1
I t a k t 1 e

    


 
  1
t
1
a k t 1 e 
     , 1
t t T
  …(2.9)
(0, T) becomes
1
2
C


 
   
1
1
t
t t
1
a k t 1 e dt
 

   

 
 
1
t
1
2
C
a k t 1 dt

    
 
Solving all integration in the above equation separately now
first integration term
 
   
1
t t
1
1
2
0
C
I a k t 1 e dt

 
    
 
 
  1
t t
1
1
2
0
C
a k t 1 e e dt

 
    
 
 
 
1
t
1
1
2
C e
a k t 1 1 e


 
     
 


 
   
1
1 t
t
1
1
3
C
I a k t 1 e e
 

 
    
 

… (p)
Now second integration term
   
t
1
3
0
C C
II 2 e dt

 
  
 
  t
1
3
0
C C
2 e e dt

 
   
 
  t
1
3 0
C C e
2 e
 

 
   
 


  
1
4
C C
II 2 1 e
    

… (q)
Now third integration term
 
          

C
a b t 1 C t 2 t 2 dt
0
2 2 2
1
3
   
 
 
  

 
C C
2 e dt
0
1
3
 
t
 
 
    

 
C
C a k t 1 e dt
2
0
1
H 1
t t
 
   
1
 
 
 
 
 
 


C C I t dt I t dt
H 1
0
t1
   
c during the period
H
Hence the inventory holding cost  
C t 2 t 2 dt
    
2 2
  
 
     

C
III a b t 1
0
1
3
2
 


2
2 2
1
3
C t
a t b b t
2

     

 
3 2
2
0
t t
C 2 C 2Ct
3 2


      

2 3
2 2 2
1
3
C
a b b C
2 3
  
        

 
2 2
2 2
2 C 2C a b b
2 2
 
          
3 2
2
C 2 C 2C
3 2

 
      

2
1 2
a b b
III C
2
   

  

  


3 2
2 3
C C 2C
3

  
   
   

… (r)
Now the fourth integration term
 
   
1
1
t
t t
1 1
2
1
IV C a k t 1 e dt
 



    




 
 
1
t
2
1
a k t 1 dt

    
 
 
 
1
1
t
t t
1
1
2
C
a k t 1 e e dt
 

    
 
 
1 1
t t
1 1
2
C C k
a k dt dt
 
   

  
 
 
1
1
t
t
1
1
2
C e
a k t 1 e e



 
     
 


  
1
1
2
C
a k t
   

2 2
1
1
C k
t
2
 
 
 

 
   
 
1
t
1
1
3
C
a k t 1 e 1
 
    

  
1
1
2
C
a k t
    

 
2 2
1
1
C k
t
2


 
   
 
1
t
1
1
3
C
a k t 1 e 1
 
    

 
1
1
2
C
a k t
  

1
2
C


 
2
a b c
    
 
2 2
1
1
C k
t
2
 

 
   
 
1
t
1
1
3
C
IV a k t 1 e 1
 
    

 
1 1 1
1
2 2
C ac C b
a k t
 
    

 
 
2
2 2
1 1
1
2
C C C k
t
2

  


… (s)
After adding p, q, r and s, we get H
C as
 
   
 
1
1 t
t
1
H 1
3
C
C a k t 1 e e
 

    

  
1
4
C C
2 1 e
   

2 3 2
1 2 2 3
a b b c c 2c
C
2 3
 
     
     
 
  
  
 
 
 
1
1
3
C
a k t 1
    

 
 
1
t
e 1
 

 
1 1
1
2
C c a
a k t

   



 
2
2 2
1 1 1
1
2 2
c b c c c k
t
2
 
   

 
 
  
1
t
1
1
3
C
a k t 1 e 1

     

  
2 3
1 1 1
4
c c c b cc
2 1 e
2 3
  
     
 

   
2 2
1 1 1
1 1
3 2
2c c c c k
a k t t
2

    

 
 
  
1
t
1
1
3
C
a k t 1 e 1

     

  
2
1 1
4
c c c b
2 1 e
2
 
    


1
3
c
3


   
3 2 1
1
2
c
c 6c a k t
      

 
2 2
1
1
c k
t
2
 

 
  
1
t
1
1
3
C
a k t 1 e 1

     

  
2
1 1
4
c c c b
2 1 e
2
 
    


1
3
c c
3



   
2 2 1
1
2
c
6 a k t
     

 
2 2
1
1
c k
t
2
 

  
H 1 4
c
C C 2 1 e

   


 
  
1
t
1
3
1
a k t 1 e 1

     

2
b
2

 

 
2 2
3
c
6
3

  

   
2 2
1 1
2
1 k
a k t t
2

     

 
…(2.10)
The cost due to deterioration of units  
D
C during the
period (0, T)
   
1
t
D 2
0
C C I t dt I t dt


 
 
   
 
 
 
   
1
t
2
0
C I t dt I t dt


 
 
  
 
 
 
  
D 2 4
c
C C 2 1 e

    


 
  
1
t
1
3
1
a k t 1 e 1

     

2
b
2



 
2 2
3
c
6
3

   

   
2 2
1 1
2
1 k
a k t t
2

     

 
…(2.11)
The cost due to shortage of units  
s
C during the given
period is given by
 
1
T
s 3
t
C C I t dt
  
 
 
1
T
t
3
s 2
t
C
C a k t 1 e

     

 
 
  1
t
1
a k t 1 e dt
 
       
 
3
1
2
C
a k t 1
    

1 1
T T
t t
3 3
2 2
t t
C C k
a e dt t e dt
 

    
 
 
1
T
t
3
2
t
C k
e dt


 
 
 
3
1
2
C
a k t 1
    

1
1
T
t
t
e dt

 
 

 
 

1 1
T T
t t
3 3
t t
C a C k
e dt t e dt
 
   
 
 
1
T
t
3
2
t
C k
e dt


 

 
 
3
1
2
C
a k t 1
    

1
1
T
t
t
e dt

 
 

 
 

1 1
T T
t t
3
3 2
t t
C k
a k
C e dt t e dt
 
 
    
 
 

 
 
 
   
1
t
3
1 1
2
C
a k t 1 e T t

     

 
1
t
T
3
2
C a k
e e

 
  
 
  
 
 
1
t
T
3
1
2
C k
T e t e

  

 
1
t
T
3
3
C k
e e

 

 
   
1
t
3
1 1
2
C
a k t 1 e T t

      

 
T
3 3 3
3
e
C a 2C k C Tk

    

 
1
t
3 3 3 1
3
e
C a 2C k C t k

    

3
2
C


 
   
1
t
1 1
a k t 1 e T t

    
 
 
T
3
s 3
C
C e a k 2 T


    


 
 
1
t
1
e a k 2 t

    
 
   
1
t
2
1 1
a k t 1 e T t

       …(2.12)
The opportunity cost due to lost sales  
0
C is given by
  
1
T
t
0 4
t
C C 1 e a kt dt

  

   
1 1
T T
t
4 4
t t
C a kt dt C e a kt dt

   
 
 
1 1
T T
t
4 4
t t
C a kt dt C ae dt

  
 
1
T
t
4
t
C k t e dt

 

 1 1 1
T T
T 2 t
4 4
4 t t t
C k C a
C at t e
2

   
  
   

1 1
T T
t t
4 4
2
t t
C k C k
te e
 
   
 
   
 
   
2 2
4
4 1 1
C k
C a T t T t
2
   
T 4 4 4
2
C a TC k C k
e  
  
 
  
 
1
t
e
 4 1 4 4
2
C a t C k C k
 
 
 
  
 
   
2 2
4
4 1 1
C k
C a T t T t
2
   
 
 
T
4
2
C e
a k 1 T

    

1
t
4
2
C e


 
 
1
a k 1 t
   
   
2 2
0 4 1 1
k
C C a T t T t
2

   


 
 
T
2
e
a k 1 T

    

1
t
2
e


 
 
1
a k 1 t 
     …(2.13)
The total average cost (k) of the inventory system permit
time is given by

 
H D s 0
C C C C C R
K
T T
    
 
where R is the total cost of ht inventory cycle
  
1
4
1 C C
K C 2 1 e
T

 
    
 

 
  
1
2
t
1 1
1
3
C c b
a k t 1 e 1
2
 
      


   
2 2
1 1
1
3 2
CC C
6 a k t
3

     
 
    
2 2
1 2
1 3
C k C C
t 2 1 e
2

     
 
 
  
1
2
t
2 2
1
2
C C b
a k t 1 e 1
2
 
      

   
2 2
2 2
1
2
CC C
6 a k t
3

     


   
 

2 2 T
3
2
1 3
C
C k
t e a k 2 T
2

      

 
 
1
t
1
e a k 2 t

    
 
   
1
t
2
1 1
a k t 1 e T t

      
   
2 2
4 1 1
k
C a T t T t
2

   


 
  T
2
1
a k T 1 e
    

 
 
1
t
1
2
e
a k t 1
 


     

 

To minimize (k), the optimal values of 1
t & T can be
obtained by solving
1
k
0
t



and
k
0
T



simultaneously.
Now
  1 1
t t
1 1 1
2
1
C a k e
k C kt e
t
 
 

 
 

1
t
1 1 1 1 1 1
2 2 2
C ke C k C a C k C kt

    
 
  
1
t
2
C ae

1 1
1
t t
t
2 2
2 1
C ke C ke
C kt e
 

  
 
2 2
2 2 1
C k C k
C a C kt
   
 
 
1 1
t t
3 3 1
2
C e C t ke
a 2k
 
   


 
1
1
t
t
2
3 3
2 2
C ke C T
a k e


    
 
1 1
t t
3
3 1
C
C Tkt e e Tk
 
 

   
1 1
t t
3 3
1 2
C C
t e a k e a k
 
     
 
1 1
t t
2 3
3 1 1
C
C kt e e .2t k
 
 

 
1
t
4
4 4 1
C e
C a C kt a k

    

1 1
t t
4
4 1
C
C e kt ke
 
 

 
1
1
t
t
1 1 1
2
C C kt e
a k e


   


1 1
t t
1 1 1 1
2
2
C C a C kt
ke C ae
 
   
 

1
t
2
C ke


1
1
t
t 2
2 1 2
C ke
C kt e C a


  


1 1 1
t t t
3 3 3 1
2 1 2
C ae 2kC e C t ke
C kt
  
   
 

1
1 1
t
t t
3 3
3
2
C C Tke
ke C Tae

 
  


1 1 1
t t t
3
3 1 3 1
C
C Tkt e e Tk C t ae
   
  

1 1 1
t t t
3 3 3
1 2
C C a C
t ke e ke
  
  
  
1 1
t t
2 3
3 1 1
C
C kt e 2t ke
 
 

   
1 1
t t
4 4 1
C a 1 e c kt 1 e
 
   
1 1 1
t t t
1 1 1
1
2 2
C C k C k
ae e t e
  
  

 
1
1
t
t
1 1 1 1
2
2
C ke C a C kt
C ae


   
 

1
t
2
C k
e


1 1
t t
2
2 1
C k
C kt e e
 
 

1
1
t
t
3 3
2 2 1 2
C a 2kC e
C a C kt e


   
 
1
t
3 1
C t ke


1
1
t
t
3
3
2
C ke
C Tae


 

1
1
t
t t
3 3
3 1
C Tke C
C Tkt e Tke

 
  
 
1
t
3 1
C t ae

1
1
t
t
3 1 3
C t k C e a
e


 
 
1
1 1
t
t t
2
3 3
3 1 1
2
C ke C
C kt e 2t ke

 
  
 
 
1
t
4
C a 1 e
   
1
t
4 1
C kt 1 e .

 
1 1
t t
1 1 1 1 1 1
C ae C kt e C a C kt
 
   
   
1 1
t t
2 2 1 2
C ae C kt e C a
 
   2 1
C kt

1
t
3
C Tae
 1 1
t t
3 1 3 1
C Tkt e C t ae
 
 
  
1 1
t t
2
3 1 4 1
C kt e C a e a kt
 
   
   
1 1
t t
1 1 1
C a C kt
e 1 e 1
 
  
 
   
1 1
t t
2 1 3 1
C kt e 1 TC e a kt
 
   
    
1 1
t t
3 1 1 4 1
C t e a kt C 1 e a kt
 
   
 
1
t 1 1 1
2 2 1
C a C kt
e 1 C a C kt
  
    
 
 
 
  
1
t
3 1 1
C e t T a kt

     
1
t
4 1
C a kt 1 e
  
     
1
t 1
1 2 1
C
e 1 a kt C a kt
  
     
 

 
  
1
t
3 1 1
C e t T a kt

     
1
t
4 1
C a kt 1 e
  
   
1 1
t t
1
2 3 1
C
e C C e t T
 
  
   
 
 
 

 
1
t
4 4 1
C e C a kt
 
  

Now putting
1
k
0
t



, we get
 
1
t 1
2
1
k C
0 e 1 C
t

   
   
 

 
 

   
1 1 4
t t C
3 1 4 1
C e t T C e a kt
   
   

 
1
t 1
2
C
e 1 C
  
 
 

 

 
 
1
t
3 1 4 4
e C t T C C 0

     …(2.14)
Now for
k
0
T



Since
R
k
T

 2
k R R 1 R
T T T T T
T
  
   
   
   
  
   
 
T
3
2 2
C e
k R 1
a 2k
T T
T



     

 

  1
t
T T
3 3 3
2 2
C kT C C
e ke a k e
 
    
  
1
t
3
1 4 4
C
t ke C a C kT

  

T
4
C
e


  T T
4
4
C
a k C kTe e k
  
    
 
T
3
2 2 2
R 1 a 2k kT k
C e
T
T
  

      
 
  
 
 
1
t 1
3 2
a k t k
C e  
  
 
 

 
  T
4 4
k k
C a kT C e a kT
 
 
      
 
 
 
T
3
2 2
R 1 a k kT
C e
T
T

  
     
 
  

 

1
t 1
3 4
2
a k t k
C e C
  
   
 
 

 
 
T
4
C e a kT
 
   
 
 
 1
t
3
1
2 2
C
R 1
e a k 1 t
T
T


      
 

 
 
T
e a k 1 T

       
T
4
C a kT 1 e 
  

Putting
k
0
T



, we get
 
 
 1
t
3
1
2 2
C
k R 1
e a k 1 t
T T
T

 
      

 

 
 
T
e a k 1 T

    
4
C
   
T
a kT 1 c 0
 
  

 
  1
t
3
1
2
C
a k t 1 e
    


 
   
T
4
a k T 1 e C a kT
 
      

 
T R
1 e 0
T

   …(2.15)
The above equations
1
k
0
t



and
k
0
T



provided, they
satisfy the following conditions :
2 2
2 2
1
2
2 2 2
2 2
1
1
k k
0, 0
t T
k k k
0
t T
t T

 
  
  

    
   
 
     
   
 
   
  
…(2.16)
Equation (2.14) and (2.15) are highly nonlinear equations.
Therefore numerical solution of these equation obtained by
using the software MATLAB 7.0.1.
4. CONCLUSIONS
This research work develops an inventory model withprice,
stock, and time-dependent demand. The physical
deterioration and condition of freshness degradation over
time are both considered, and zero-ending inventory is
assumed. When working with perishable products, a
salvaged value and a deterioration costareconsideredinthe
entire cycle. A nonlinear time-dependent holding cost is
included, specifically with a quadratic-type function.

Through an algorithm, the inventory model determines the
optimal values for price, the inventory cycle time, and the
order quantity. Some numerical examples are provided,and
a sensitivity analysis is presented for all the input
parameters. By observing the behavior of the decision
variables and total profits, it was found that an increase in
the ordering cost , purchasingcost , andshelf-life results ina
similar pattern in the selling price, the inventory cycle time,
the quantity to order, and the total profit. Furthermore, an
increase in the value of the shelf-life results in an increment
in price, inventory cycle time, quantity ordered, and profits
generated for all functions. In addition, as the ordering cost
increases, price, the inventory cycle time and quantity
ordered also increase for all functions. Nonetheless, the
profits show a decreasing trend. Finally, by escalating the
purchasing cost for all functions, there is an increase in both
the price and the inventory cycletime;however,thequantity
to order and total profits tend to decrease.
This research work extends and widely contributes to the
state-of-the-art on the inventory field, with focus on
perishable items with price-stock-time-dependent demand.
The inventory model studied here hassomelimitationsfrom
where several directions for extension and further research
are highlighted. First, an inventory model can be built with
the same characteristics and demand pattern, but including
the sustainability elements, so the effects of the carbon-tax
and cap-and-trade mechanisms can be assessed. Second, a
model that allows shortages with full or partial backlogging
should be explored. Third, the trade-off and benefits of
investing on preservation technologyshouldbealsostudied.
Fourth, the noninstantaneous item’s freshness degradation
can be integrated intotheproposedinventorymodel.Finally,
other components such as incorporatingdiscountpoliciesor
advertising efforts can also be investigated.
5. REFERENCES
[1] A. H. M. Mashud, H. M. Wee, C. V. Huang, and J. Z. Wu,
“Optimal replenishment policy for deteriorating
products in a newsboy problem with multiple just-in-
time deliveries,” Mathematics, vol. 8, no. 11, pp. 1–18,
2020.
[2] A. H. M. Mashud, D. Roy, Y. Daryanto, and M. H. Ali, “A
sustainable inventory model with imperfect products,
deterioration,andcontrollableemissions,” Mathematics,
vol. 8, no. 11, pp. 1–21, 2020.
[3] B. Pal, S. S. Sana, and K. Chaudhuri, “A stochastic
production inventorymodel fordeterioratingitemswith
productsʼ finite life-cycle,” RAIRO-Operations Research,
vol. 51, no. 3, pp. 669–684, 2017.
[4] Bhunia A.K and M. Maiti, Deterministic inventory model
for deteriorating items with finite rate of replenishment
dependent on inventory level. Computers and
Operations Research 25 (11), 997-1006, 1998.
[5] Cheng TCE., An economic order quantity model with
demand-dependent unit production cost and imperfect
production processes, IIE Transactions 23, 23-28,1991.
[6] Chu. P and Patrick S.H., A note on inventory
replenishment policies for deteriorating items in an
exponentially declining market. Computers and
Operations Research 29 (13), 1827-1842, 2002.
[7] Covert, R.P and G. C. Philip, An EOQ model foritems with
Weibull distributed deterioration. AIIE Transactions 5,
323-326, 1973.
[8] E. B. Tirkolaee, A. Goli, M. Bakhsi, and I. Mahdavi, “A
robust multi-trip vehicle routing problem of perishable
products with intermediate depots and time
windows,” Numerical Algebra, Control & Optimization,
vol. 7, no. 4, pp. 417–433, 2017.
[9] Giri, B.C. and Chaudhari K.S., “Deterministic models of
perishable inventory withstock dependentdemandrate
and nonlinear holding cost”, European Journal of
Operational Research, 105, 3, pp467-474, 1998.
[10] Yan H. and TCE Cheng, Optimal productionstopping and
restarting times for an EOQ model with deteriorating
items. Journal ofOperational ResearchSociety,49, 1288-
1295, 1998.
[11] Kuo-Lung Hou, An inventory model for deteriorating
items with stock-dependent consumption rate and
shortages under inflation and time discounting.
European Journal of Operations Research 168,463-474,
2006.
[12] Chandra M.G., An EPQinventorymodel for exponentially
deteriorating items under retail partial trade policy in
supply chain, Expert Systems and applications. 39(3),
3537-3550, 2012.
[13] M. Yavari, H. Enjavi, and M. Geraeli, “Demand
management to cope with routes disruptions in
location-inventory-routing problem for perishable
products,” Research in Transportation Business &
Management, vol. 37, Article ID 100552, pp. 1–14,2020.
[14] Teng J.T. and Chang C.T., Economic production quantity
models for deteriorating items with price- and stock-
dependent demand. Computers and Operations
Research 32, 297-308, 2005.
[15] Zaid T. Balkhi & Lakdere Benkherouf, An inventory
model for deteriorating items with stock dependent
demand and time-varyingdemandrates.Computersand
pertains Research 31, 223-240, 2004.

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