An inventory model for perishable items with quadratic trapezoidal type demand under partial backlogging

Article history: Received March 9 2014 Received in Revised Format August 5 2014 Accepted December 3 2014 Available online December 3 2014 In this paper, we consider the inventory model for perishable items with quadratic trapezoidal type demand rate, that is, the demand rate is a piecewise quadratic function under constant deterioration rate. The model consider allows for shortages and the demand is partially backlogged. The model is solved analytically by minimizing the total inventory cost. The result is illustrated with numerical example. Finally, we discuss sensitivity analysis for the model. © 2015 Growing Science Ltd. All rights reserved


Introduction
Deteriorating items are very common issue in our daily life circumstances.In recent years, many researchers have studied inventory models for deteriorating items, however, academia has not reached a consensus on the definition of the deteriorating items.According to Wee (1993), deteriorating items refers to the items that become decayed, damaged, evaporative, expired, invalid, devaluation and so on through time.According to the definition, deteriorating items can be classified into two categories.The first category refers to the items that become decayed, damaged, evaporative, or expired through time, like meat, vegetables, fruit, medicine, flowers and so on; the other category refers to the items that lose part or total value through time because of new technology or the introduction of alternatives, like computer chips, mobile phones, fashion and seasonal goods and so on.The inventory problem of deteriorating items was first studied by Whitin (1957), he studied fashion items deteriorating at the end of the storage period.Then, Ghare and Schrader (1963) concluded in their study that the consumption of the deteriorating items was closely relative to a negative exponential function of time.Various authors such as Deng et al. (2007), Cheng and Wang (2009), Cheng et al. (2011) and Hung (2011) studied inventory models for deteriorating items in various aspects.In world business market, demand has been always one of the most key factors in the decisions relating to the inventory and production activities.There are mainly two categories demands in the present studies, one is deterministic demand and the other is stochastic demand.Various formations of consumption tendency have been studied, such as constant demand (Padmanabhan & Vrat, 1990;Chung & Lin, 2001;Benkherouf et al., 2003;Chu et al., 2004), level-dependent demand (Giri & Choudhuri, 1998;Chung et al., 2000;Bhattacharya, 2005;Wu et al., 2006), price dependent demand (Wee & Law, 1999;Abad, 1996Abad, , 2001)), time dependent demand (Resh et al., 1976;Henery, 1979;Sachan, 1984;Dave, 1989;Teng, 1996;Teng et al., 2002;Skouri & Papachristos, 2002;Panda et al., 2012;Sett et al., 2013;Mishra et al., 2013) and time and price dependent demand (Wee, 1995).Among them, ramp type demand is a special type of time dependent demand.Hill (1995), one of the pioneers, developed an inventory model with ramp type demand that begins with a linear increasing demand until to the turning point, denoted as  , proposed by previous researchers, then it becomes a constant demand.There has been a movement towards developing this type of inventory system for minimum cost and maximum profit problems.Several authors: Mandal and Pal (1998) focused on deteriorating items.Wu et al. (1999) were concerned with backlog rates relative to the waiting time.Wu and Ouyang (2000) tried to build an inventory system under two replenishment policies: starting with shortage or without shortage.Wu (2001) considered the deteriorated items satisfying Weibull distribution.Giri et al (2003) dealt with more generalized three parameter Weibull deterioration distribution.Deng (2005) extended the inventory model of Wu et al. (1999) for the situation where the in-stock period is shorter than  .Manna and  Chaudhuri (2006) set up a model where the deterioration is dependent on time.Panda et al. (2007) constructed an inventory model with a comprehensive ramp type demand.Deng et al. (2007) contributed to the revision of Mandal and Pal (1998), and Wu and Ouyang (2000).Panda et al. (2008) examined the cyclic deterioration items.Wu et al. (2008) studied the maximum profit problem with the stockdependent selling rate.They developed two inventory models all related to the conversion of the ramp type demand, and then examined the optimal solution for each case.However, in a realistic product life cycle, demand is increasing with time during the growth phase.Then, after reaching its peak, the demand becomes stable for a finite time period called the maturity phase.Thereafter, the demand starts decreasing with time and eventually reaching zero or constant.
In this work, we extend Hill's ramp type demand rate to quadratic trapezoidal type demand rate.Such type of demand pattern is generally seen in the case of any fad or seasonal goods coming to market.The demand rate for such items increases quadratic-ally with the time up to certain time and then ultimately stabilizes and becomes constant, and finally the demand rate approximately decreases to a constant, and then begins the next replenishment cycle.We think that such type of demand rate is quite natural and useful in real world market situation.One can think that our work may provide a solid foundation for the future study of this kind of important inventory models with quadratic trapezoidal type demand rate.

Assumption and notations
The fundamental assumption and notations used in this paper are given as follows: (1) The demand rate, R(t), which is positive and consecutive, is assumed to be a quadratic trapezoidal type function of time, that is Chose a1,b1,c1,a2,b2


. where 1  is the time point changing from the increasing quadratic demand to constant demand, and 2  is the time point changing from the constant demand to the decreasing demand.
(3) I(t) is the inventory level at any time t, T t   0 .(4) T is the fixed length of each ordering cycle.
(5)  is the constant rate of deterioration, ) t1 is the time when the inventory level reaches zero.(7) t1 * is an optimal point.(8) k0 is the fixed ordering cost per order.(9) k1 is the cost of each deteriorated item.(10) k2 is the inventory holding cost per unit per unit of time.
(11) k3 is the shortage cost per unit per unit of time.
(12) S is the maximum inventory level for the ordering cycle, such that S=I(0).(13) Q is the ordering quantity per cycle.( 14) A1(t1) is the average total cost per unit time under the condition

Mathematical and theoretical results
Here, we consider the deteriorating inventory model where demand rate is trapezoidal type quadratic function.Replenishment occurs at time t =0 when the inventory level attains its maximum.For ] , 0 [ 1 t t  , the inventory level reduces due to both demand and deterioration.At time t1, the inventory level reaches zero, then shortage is allowed to occur during the interval (t1, T), and all of the demand during the shortage period (t1, T) is completely backlogged.The total amount of backlogged items is replaced by the next replenishment.The rate of change of the inventory during the stock period [0, t1] and shortage period (t1, T) is governed by the following differential equations: with boundary condition I(0)=S and I(t1)=0.One can think about t1, t1 may occur within ] , 0 [ . Hence in this paper we are going to discuss all three possible cases.
Case 1: The quadratic trapezoidal type market demand and constant rate of deterioration, the inventory level gradually diminishes during the period [0, t1] and ultimately reaches to zero at time t=t1.Then, from Eq.
(2) and Eq.(3), we have Now solving the differential Eqs.(4-7) with the condition I(t1)=0 and continuous property of I(t), we get The beginning inventory level can be computed as The total number of items which is perish in the interval [0, t1], say DT, is The total amounts of inventory carried during the interval [0, t1], say CT, is The total shortage quantity during the interval [t1, T], say BT, is The average total cost per unit time for The first order derivative of ) ( 1 1 t A with respect to 1 t is as follows: t1) is a strictly monotonically increasing function and Eq. ( 19) has a unique solution at * 1 t , for ) , 0 ( * 1

T t 
. Therefore, we have

Property-1
The constant deteriorating rate of an inventory model with quadratic trapezoidal type demand rate under the time interval The total back order amount at the end of the cycle is follows, Therefore, the optimal order quantity, denoted by * Q , is , where * S denote the optimal value of S.

Case-II,
, then, the differential equations governing the inventory model can be expressed as follows: Solving differential Eq. (22-25), using I(t1)=0, we get The beginning inventory can be computed as The total amount of items which is perish within the time interval [0, t1] is The total amount of inventory carried during the time interval [0, t1] is The total amount of shortage during the interval [t1, T] Now, the average total cost per unit time under the condition The first order derivative of ) ( 1 2 t A with respect to 1 t is given by The required necessary condition for

Property-2
The constant deteriorating rate of an inventory model with quadratic trapezoidal type demand function during the time interval Now, we can calculate the total amount of back-order quantity at the end of the cycle is Therefore, the optimal order quantity denoted by * Q is , where * S denotes the optimal vale of S.

Case-III
, then, the differential equations governing the inventory model can be expressed as follows: Solving the differential Eqs.(39-42) with I(t1)=0, we can get The total amount of inventory level at the beginning can be computed as The total amount of items which is perish within the time interval [0, t1] is The total amount of inventory carried during the time interval [0, t1] is Total quantity of shortage during the time interval [t1, T] is Then, the total average cost per unit time under the time interval The first order derivative of ) ( 1 3 t A with respect to 1 t is as follows: The required necessary condition for

Property-3
In this case, the inventory model under the condition attains its minimum at . Now, we can calculate the total back-order quantity at the end of the cycle is Therefore, the optimal order quantity, denoted by * , where * S denotes the optimal value of S. From the above three cases, we can derive the following results

Result-1
An inventory model having constant deteriorating rate with quadratic trapezoidal type demand, the optimal replenishment time is In the above table some sensitivity analysis of the model is performed by changing the parameter -50%, -25%, -20%, -10%, +10%, +20%, +25% and +50% taking one at a time and keeping the remaining unchanged.

Conclusion
In a realistic product life cycle, demand is increasing with time during the growth phase.Then, after reaching its peak, the demand becomes stable for a finite time period called the maturity phase.Thereafter, the demand starts decreasing with time.Therefore, in this paper, we have studied the inventory model for constant deteriorating items with quadratic trapezoidal demand rate.We have proposed an inventory replenishment policy for this type of inventory model.From the market information, we have found that the quadratic trapezoidal type demand rate was more realistic than ramp type demand rate, constant demand rate and other time dependent demand rate Our paper provides an interesting topic for the future study of such kind of important inventory models, and at the same time, the following problems can be considered for future research work (1) How about the inventory model starting with shortages?(2) How about the inventory model with time dependent deteriorating rate instead of constant deteriorating rate?
and c2 such a way that