An optimization of an inventory model of decaying-lot depleted by declining market demand and extended with discretely variable holding costs

Article history: Received July 2 2013 Received in revised format September 7 2013 Accepted September 15 2013 Available online September 2


Introduction
One of the most important concerns of inventory management is to decide when and how much to order so that the total cost associated with the inventory system can be kept at minimum level.When inventory is decaying in nature, it becomes more important since deterioration cannot be ignored.There are various studies in this direction in continuous modification of inventory model for decaying items by including more and more practical features.Researchers are engaging in analyzing inventory models for deteriorating items such as volatile liquids, medicines, electronic components, fashion goods, fruits, vegetables, etc.An order level inventory model with constant deterioration was first developed by Aggarwal (1978).Now, the inclusion of deterioration aspect into the inventory concept is incorporated in wide range of considered business environments in contemporary inventory models.Sana (2010) studied optimal selling price and lot size with time varying deterioration and partial backlogging.In this effort, an EOQ model over an infinite time horizon for perishable item where demand is price reliant and partial backorder permitted is discussed.Liao and Huang (2010) developed a deterministic inventory model for deteriorating items with trade credit financing and capacity constraints.They offered an inventory model for optimizing the replenishment cycle time for a single deteriorating item under a permissible delay in payments and constraints on warehouse capacity.Hung (2011) urbanized an inventory model with generalized type demand, deterioration and backorder rates.Bhunia and Shaikh (2011) developed a deterministic model for deteriorating items with displayed inventory level dependent demand rate incorporating marketing decisions with transportation cost.Khanra et al. (2011) offered an EOQ model for a deteriorating item with time-dependent quadratic demand under permissible delay in payment.In this study, a step was taken to analyze an EOQ model for deteriorating item considering quadratic time dependent demand rate and permissible delay in payment.
In various situations of inventory control, demand before ending spell exists and the inventory has mostly consumed through joint effect of the demand and the deterioration.This type of situations laid the foundation of supply out phenomena.Consequently, when supply out state occurs, some clients are willing to wait for backorder and others may wish to buy from supplementary sellers.Many researchers such as Park (1982), Hollier and Mak (1983) and Wee (1995) well thought-out the constant partial backlogging rates during the shortage period in their inventory models.In most inventory systems, the length of the waiting time for the next replenishment would come to a decision whether the backlogging will be accepted or not.Therefore, the backlogging rate is variable and dependent on the waiting time for the next replenishment.Chang and Dye (1999) investigated an EOQ model allowing shortage and partial backlogging.They assumed in their inventory model that the backlogging rate was variable and dependent on the length of the waiting time for the next replenishment.Many researchers modified inventory policies by considering the ''time-proportional partial backlogging rate'' such as Abad (2000), Papachristos and Skouri (2000), Wang (2002), Papachristos and Skouri (2003), etc. Teng et al. (2003) then unmitigated the fraction of unsatisfied demand back ordered to any decreasing function of the waiting time up to the next replenishment.Teng and Yang (2004) widespread the partial backlogging EOQ model to allow for time-varying purchase cost.Yang (2005) prepared a comparison among various partial backlogging inventory lot size models for deteriorating stuffs on the basis of maximum profit.Teng et al. (2007) compared two pricing and lot sizing model for deteriorating objects with shortages.Dye et al. (2007) urbanized inventory and pricing strategies for deteriorating items with shortages.Skouri et al. (2011) projected an inventory model with general ramp type demand rate, constant deterioration rate, partial backlogging of unfulfilled demand and conditions of permissible delay in payments.Other related articles on inventory system with partial backlogging and shortages have been performed by Hou (2006), Jaggi et al. (2006Jaggi et al. ( , 2012)), Patra et al. (2010), Yang et al. (2010), Lin (2012), Taleizadeh et al. (2011Taleizadeh et al. ( , 2012)), etc.However, a few number of researchers paid their attention towards generalizing the term of holding cost into the inventory models.Therefore, there are few literatures of inventory controlling phenomena under the aspect of variable holding cost.As alarmed above, most researchers unspecified that holding cost rate per unit time is invariable.However, more sophisticated storeroom facilities and services may be required for holding perishable items if they are kept for longer time.Therefore, in holding of perishable items, the assumption of unvarying holding cost rate is not always apt.Weiss (1982) noted that variable holding costs are suitable when the value of an item decreases the longer it is in stock.Ferguson et al. (2007) indicated that this type of model is suitable for perishable items in which price markdowns or removal of aging product are necessary.Alfares (2007) also assumed an inventory model with discretely variable holding cost.Recently, Mishra and Singh (2011) developed the inventory model for deteriorating items with time dependent linear demand and holding cost.
To give attention on the concept of variability of the holding cost of decaying item, Tyagi et al. (2012) developed an inventory model for decaying item with power demand pattern and managed first Weibull function for holding cost rate.In that study, the holding cost depends continuously on deterioration cost and storage period, shortages were allowed and partially backlogged inversely with the waiting time for the next replenishment.Therefore, this study has left a clear vacuum for study of the discrete change in the holding cost under considering environment of inventory set-ups.Tripathi (2013) studied an inventory model for time varying demand and constant demand; and time dependent holding cost and constant holding cost for case 1 and case2 respectively.He considered non-decaying items in his model and give a motivation to study our model for deteriorating items with discrete holding cost.
In result, an Economic Order Quantity (EOQ) inventory model of deteriorating item is considered with continuosly declining market demand.To extend such EOQ model in above mentioned directions, it is assumed that the holding cost rate per unit per unit time is discrete variable with respect to time and the deterioration rate of item is considered as two-parameter Weibull distributive function.Partial backlogging is allowed.The backlogging rate is an exponentially decreasing function of the waiting time for the next replenishment.
In this study, the primary problem is to minimize the average total cost per unit time by optimizing the shortage point per cycle.Separateing for each scenario, we show that minimized objective function is convex and the optimal solution is uniquely determined.Numerical example is proposed to illustrate the model and the solution procedure for each scenario of holding cost.The sensitivity analysis of major parameters is separately performed.

Notations
The following notations are used throughout the whole chapter

Assumptions
In developing the mathematical model of the inventory system, the following assumptions are made: 1


, where 0   , and t is the waiting time.

Model Formulations
As depicted above, the inventory arrangement goes like this: At 0 t  , opening replenishment Q units are made, in which S units are delivered towards backorders, leaving a balance of max I units in the initial inventory.From 0 t  to 1 t t  time units, the inventory level depletes owing to both demand and deterioration.At 1 t , the inventory level is zero.During the time 1 ( ) T t  part of the shortage is backlogged and part of it is lost sales.Only the backlogging items are replaced by the after that replenishment.

Fig. 1. Inventory system of decaying item for declining market demand
The inventory function with respect to time can be determined by evaluating the differential equations And with boundary conditions max (0) I I  and 1 ( ) 0. I t  The approximate solution of Eq. ( 1) by neglecting higher order term of is (3) Now, again taking the first two terms of the exponential series and neglecting the terms containing 2  Eq. ( 4) becomes So, the maximum inventory level for each cycle can be obtained as During the shortage interval   1 , t T , the demand at time t is partially backlogged at the fraction ( ) Thus, the solution of differential Eq. ( 2) governing the amount of demand backlogged is as below with the boundary condition 1 ( ) 0 I t  .Let t T  in Eq. ( 6), we obtain the maximum amount of demand backlogged per cycle as follows. 1 (7) Hence, the order quantity per cycle is given by 1 2 (1 ) (1 ) The order cost per cycle is The deterioration cost per cycle is The shortage cost per cycle is The opportunity cost per cycle is

Holding Cost
Holding of inventory is a central part of inventory controlling phenomena.When item in collection has a deteriorating nature, it is more to be concerned of such items in stock holding.The owners of inventory have to endow not only for holding such item's units but also invest in handling these items for guardianship in good conditions.We are fascinated by this aspect to demonstrate a mathematical inventory model that can give us a picture which is better and very near to realities of business upbringing.Therefore, here we have understood that the holding cost of inventory is not constant and always depends upon time for which it has held.Now, here holding cost is measured as discretely variable holding cost with storage period.For using these assumptions, we have considered first two scenarios for discrete nature of variability of holding cost as retroactively variable holding cost and incrementally variable holding cost as: Scenario 1: Retroactive holding cost; Scenario 2: Incremental holding cost;

Scenario 1: Retroactive Holding Cost
In this scenario, the unit holding cost per unit time is well thought-out as discrete in nature, and increases as the time in storage increases, 1 2 3 ... n h h h h     , for storage periods 1 through n, respectively.A retroactive holding cost implies that the holding cost of the last storage period is applied retroactively to all previous periods in the order cycle.That is, if the cycle length is 1  or less, the unit holding cost is 1 h per time period; if the cycle length is between 1 is charged a holding cost of 2 h per unit per time period; etc.Since the same holding cost will be applied to all units in the cycle, we only need to determine the total inventory level for the entire order cycle: 1 0 ( ) where h is the corresponding value of In the first scenario, the objective is to determine the optimal values of shortage point 1 t in order to minimize the average total cost 1 1 ( ) ATC t per unit time.The optimal solutions * 1 t need to satisfy the following equation. where 1   then the solutions to Eq. ( 15) not only exists but also is unique (i.e., the optimal values * 1 t is uniquely determined).
Proof: From (15), it is easily verified that, when . It implies that the ( 15) is verified at and  1   the average total cost per unit time Proof: From Eq. ( 15 ( ) ATC t .This completes the proof.
In this scenario, by using * 1 t , we can obtain the optimal maximum inventory level and the minimum average total cost per unit time from Eq. ( 5) and Eq. ( 14), respectively (we denote these values by max I and * 1 1 ( ) ATC t ).Furthermore, we can also obtain the optimal order quantity (we denote it by * Q ) from Eq. (8).

Scenario 2: Incremental Holding Cost
In this scenario, the discrete incremental unit holding cost increases as the time in storage increases.In this situation, though, an incremental holding cost implies that the holding cost of each storage period is applied only to the units apprehended during that period.That is, if the positive inventory time length is 1  or less, the unit holding cost is 1 h per time period; if the storage time-span is between 1 1 2 t     , the holding cost of 1 h is applied to the average inventory during the storage period from 0 to 1  and 2 h is applied from 1  to 1 t ; etc. Thus, we require evaluating the average inventory level for each storage phase within the order cycle (note, for the last storage period, i . Thus, the average total cost 2 1 ( ) T t e c D T t .
In this scenario, the objective is to determine the optimal values of shortage point 1 t in order to minimize the average total cost 2 1 ( ) ATC t per unit time.The optimal solutions * 1 t need to satisfy the following equation. where , then the solutions to Eq. ( 19) not only exists but also is unique (i.e., the optimal values * 1 t is uniquely determined).
Proof: From Eq. ( 19), it is easily verified that, when Proof: From Eq. ( 19), if ATC t .This completes the proof.In this scenario, by using * 1 t , we can obtain the optimal maximum inventory level and the minimum average total cost per unit time * 2 1 ( ) ATC t from ( 5) and ( 19), respectively.Furthermore, we can also obtain the optimal order quantity from (8).

Numerical Examples
As an illustration of both scenarios of developed model, a numerical example is presented for a single product.To perform the numerical analysis, data have been taken randomly from literatures in appropriate units.
Example 1: We consider an inventory system which verifies the described assumptions above.The input data of parameters are taken randomly as and 4 2 c  .
By using MATHEMATICA 8.0, the global minimum Average Total Cost per unit time 1 ( ) along with the optimal value of * 1 t is calculated for each the proposed i-th scenario.The Optimal Order Quantity * ( ) Q is also calculated in each scenario.The summary of crucial values for each scenario is given below.Observations: One can make following remarks.i.The Optimal Average Total Cost per unit time is greater in the scenario 1.
ii.The Optimal Order Quantity has maximum value in the scenario 2. Optimal oreder quantity Average total cost per unit time

Sensitivity Analysis
In this section, the effects of studying the changes in the optimal value of Average Total Cost per unit time, the optimal shortage point and the optimal value of Order Quantity per cycle of each scenario with respect to changes in some model parameters are discussed.The sensitivity analysis in each scenario is performed by changing the value of each of the parameters by 5%  and 10%  , taking one parameter at a time and keeping the remaining parameters unchanged.Example 1 is used in each scenario.

Sensitivity Analysis for Scenario 1
To discuss the effect of changes of model parameters 1 Q are presented in the following Table 2.

Sensitivity Analysis for Scenario 2
To discuss the effect of changes of model parameters 1 Q are presented in the following Table 3.

Conclusions
In this model, we have studied an inventory model in which the inventory is depleted not only by declining pattern of demand but also by Weibull distributed deterioration where holding cost per unit time is considered a discretely variable.Shortages are allowed and partially backlogged.Conditions for existence and uniqueness of the optimal solution have been provided.Therefore, the proposed model can be used widely in inventory-control of certain deteriorating items such as food items, electronic components, and fashionable commodities, and others.Moreover, the advantage of the proposed inventory model is illustrated with example.This study highlights that the optimal average total cost per unit time is high when holding cost per unit per unit time is considered as retroactively to all previous periods of storing and optimal value of ordered quantity is less.On the other hand, the optimal average total cost per unit time is less when holding cost per unit per unit time is considered as incremental to periods of storing and optimal value of ordered quantity is high.In future, this paper may be extended with stochastic demand and permissible delay of payment.
convex and reaches its global minimum at point * 1 t .

Fig. 2 .
Fig. 2. Inventory model optimal values for each scenario

Fig. 3 .Fig. 4 .
Fig. 3. Behavior of optimal average total cost per unit time in scenario 1

Fig. 5 .Fig. 6 .
Fig. 5. Behavior of optimal average total cost per unit time in scenario 2 Shortages are permitted.Unfulfilled demand is partially backlogged.The backlogging rate ( ) B t which is a decreasing function of the waiting time t for next replenishment, we here assume that ( ) t B t e 

Table 1
Summary of model's optimal values in i-th scenario No. of scenario

Table 2
Sensitivity Analysis for Scenario 1

Table 3
Sensitivity Analysis for Scenario 2 and  .It is less sensitive to changes in ,   and 1 c ; and very less sensitive to change in 1 h and  .