1. Introduction

In real life situations there is a realistic phenomenon of the deterioration of goods in a storage period. Deterioration is defined as damage, decay or spoilage of the items that are stored for future use always loose part of their value with passage of time, so deterioration cannot be avoided in any business organization. Covert and Philip [1973] developed an inventory model for exponentially decaying items. Buzacott [1975] developed an economic order quantity model and first time he assumed the effect of inflation in his inventory model. Donaldson [1979] determine an optimal replenishment policy for finding the computational solution of the inventory model of deteriorating items. Deb and Chaudhuri [1987] extended the Donaldson [1979] model by allowing shortages.

In classical inventory models the demand rate is either constant, linearly increasing, decreasing, exponentially increasing, decreasing function of time or stock dependent. Later it has been observed that in the super market the above demand pattern do not precisely depict the demand of certain items such as newly launched products, fashionable garments, hardware devices, cosmetics, electronic items, mobiles etc. increases with time and after some time it becomes constant. In such cases the concept of ramp-type demand is introduced. Ramp-type demand is a demand which increases up to a certain time and after a certain time it becomes constant. Dutta and Pal [1991] discussed the effect of inflation and time value of money in his inventory model by considering a linearly time dependent demand rate. Mandal and Pal [1998] were the first authors who discussed the ramp-type demand rate in his order level inventory model for deteriorating items. Chang and Dye [1999] developed an EOQ model for deteriorating items in which backlogging rate is reciprocal of a linear function of waiting time. Wu, K.S., Ouyang, L.Y. (2000) A replenishment policy for deteriorating items with ramp-type demand rate. Wu [2001] formulated an EOQ model for weibull deteriorating items with ramp-type demand rate and partial backlogging. Teng et al. [2002] determines an optimal replenishment policy for deteriorating items with time varying demand rate by allowing shortage. Manna, S.K. and Choudhuri, K.S. (2003) An EOQ model with ramp-type demand rate, time dependent deterioration rate, unit production cost and shortages. Jaggi et al. [2006] determine an optimal order policy for deteriorating items under inflation and discounted cash flow approach over a finite planning horizon. Dye et al. [2006] considered an inventory model for deteriorating items with time varying demand rate and shortage. Deng et al. (2007) A note on inventory models for deteriorating items with ramp-type demand rate. Panda et al. (2008) Optimal replenishment policy for perishable seasonal products in a season with ramp-type time dependent demand. Kun-Shan et al. (2008) developed a retailer’s optimal ordering policy for deteriorating items with ramp-type demand under stock-dependent consumption rate. Chung and Wee [2008] formulated a policy pricing integrated production inventory model for deteriorating items by considering imperfect production, inspection planning and stock dependent demand rate. Skouri et al. [2009] developed an inventory model for weibull deteriorating items with ramp-type demand rate and shortage. Cardenas Barron [2009] proposed an inventory model with a rework process at a single stage manufacturing system with planned backorders. Jain, S. and Kumar, M. (2010) formulated an EOQ model for three parameter weibull deteriorating items with ramp-type demand and shortages. Sana [2010] developed a multi item EOQ model for both deteriorating and ameliorating items. Sarkar et al. [2010] described a production policy to find out an optimal safety stock, production lot size and reliability parameters. Sana [2010] formulated a lot size inventory model for deteriorating items with time varying deterioration rate and partial backlogging. Chang, C.T. (2011) developed an inventory model for weibull deteriorating items with ramp-type demand rate and partial backlogging. Wee et al. [2011] determines an optimal replenishment cost of life analysis of deteriorating green products. Cardenas- Barron [2011] considered an inventory model with shortage and find out an approximate solution by using basic algebraic procedure. Sarkar and Moon [2011] extended the economic production quantity model in an imperfect production system. Sett et al. [2012] formulated a two warehouse inventory model for time varying deteriorating items and stock dependent demand rate. Ahmad et al. (2013) developed an inventory model with ramp-type demand rate, partial backlogging and general deterioration rate. Cardenas et al. [2013] determines an improved solution procedure of the replenishment policy for the EMQ model with rework and multiple shipments. Sarkar and Majumder [2013] developed an integrated vendor buyer supply chain inventory model with the reduction of vendors set up cost. Cardenas et al. [2013] derived two easy and improved algorithms to determine jointly the replenishment lot size and number of shipments for an EPQ model. Karmakar, B. and Choudhuri, K.D. (2014) developed an inventory model for deteriorating items with ramp-type demand rate, partial backlogging and time varying holding cost. Sarkar et al. [2014] developed an inventory model with trade credit policy and variable deterioration rate for fixed life time products. Sarkar et al. [2014] developed an EMQ model with price and time dependent demand under inflation. Sarkar et al. [2015] derived a continuous review manufacturing inventory model with set up cost reduction, quality improvement and a service level constraint. Kumar et al. (2015) developed a two warehouse partially backlogging inventory model for deteriorating items with ramp-type demand rate.

The tables 1 and 2 show that the variation of total profit when with respect to the change in deterioration parameter and inflation parameter , the tables 3 and 4 show that the variation of total profit when with respect to the change in deterioration parameter and inflation parameter . The figures 1 and 2 are correspond to the developed model in two cases and . The figures 3 and 4 show that the variation of total profit when with respect to the change in deterioration parameter and inflation parameter and the figures 5 and 6 show that the variation of total profit when with respect to the change in deterioration parameter and inflation parameter .

Figure 1. For case I

Figure 2. For case II

Figure 3. With respect to θ

Figure 4. With respect to ρ

Figure 5. With respect to θ

Figure 6. With respect to ρ

In this paper we developed a partially backlogging inventory model for deteriorating items with probabilistic deterioration rate and ramp type demand under stock dependent consumption rate. Shortages are allowed and completely backlogged, the backlogging rate of unsatisfied demand is assumed as a function of waiting time. The effect of inflation and time value of money is introduced in the model. The ramp type demand is a demand which increases up to a certain time and after that it becomes stable or constant. In the case of real estate, electronics items, fashionable garments, cosmetics, hardware devices, food grains etc. the demand is unknown with certainty while the supply is dependent on consumers need so in this paper we consider an inventory model with deterministic demand.

Although there are so many research papers related to the ramp-type demand rate of newly launched products in the super market. This paper deals with the same type problem and the purpose of our study is to provide an approximate solution procedure for an optimal replenishment policy to maximizing the total profit.

2. Assumptions and Notations

We consider the following assumptions and notations corresponding to the developed model.

1. The ramp type demand rate is where a is the initial demand rate and b a constant governing exponential demand rate and is the Heaviside unit step function of time defined by

2. The selling rate is influenced by the demand rate and on hand inventory stock by,

3. Where, k is a positive constant.

4. is the probabilistic deterioration rate.

5. is the backlogging parameter.

6. is the ordering cost per order.

7. is the holding cost per unit

8. is the shortage cost.

9. is the purchase cost per unit.

10. is the lost sell cost per unit.

11. is the selling price per unit.

12. r is the discount rate represent the time value of money.

13. i is the inflation rate per unit time.

14. is the discount rate minus inflation rate.

15. is the parameter of ramp-type demand rate.

16. is the inventory level at any time in [0,T].

17. is the time of zero inventory level.

18. T is the length of each ordering cycle.

19. Q is the order quantity per cycle.

20. The replenishment rate is infinite.

21. Shortages are allowed and partially backlogged.

22. The lead time is zero.

23. is the total profit per unit time for model I.

24. is the total profit per unit time for model II.

3. Mathematical Formulation

Suppose an inventory system consists the maximum inventory level at time t=0 in the beginning of each cycle. During the time interval the inventory level decreases due to both demand and deterioration and becomes zero at The shortage starts at and shortage interval is the end of current order cycle. During the shortage interval shortages are allowed and partially backlogged. The unsatisfied demand is backlogged at rate of , where t is the waiting time and the positive constant is the backlogging parameter.

The instantaneous inventory level at any time t in is governed by the following differential equations

(1)

with boundary condition

(2)

with boundary condition

The solution of these equations is affected by the selling rate. Now we discuss the following two cases and

Case I When then in the interval the selling rate is defined as

Using in equations 1 and 2

(3)

with boundary condition

(4)

with boundary condition

(5)

with boundary condition

The equation (3) can also be written as

For the solution of equation (3) the integrating factor is

and the solution of equation (3) is

The solution of equation (3) is

(6)

For the solution of equation (4) it can be written as

The solution of equation (4) is

(7)

For the solution of equation (5) it can be written as

The solution of equation (5) is

(8)

Using the condition and from the equation (6) we obtain

(9)

Using in the equation (6) we obtain the solution of equation (3)

(6A)

When then from the equations (7) and (8)

(10)

Using the value of q in equation (8) we obtain the solution of equation (5)

(8A)

The maximum order quantity Q is

(11)

The ordering cost per cycle is

(12)

The holding cost per cycle is

(13)

The shortage cost per cycle is

(14)

The purchase cost per cycle is

(15)

The lost sales cost per cycle is

(16)

The sales revenue per cycle is

(17)

The total profit per unit time under the effect of inflation and time value of money is

(18)

Now our objective is to determine the optimal value of for which the total profit is maximum.

The necessary condition for to be maximum is that

and solving these equations we find the optimum values of say for which profit is maximum and the sufficient condition is

(19)

(20)

(21)

(22)

(23)

Case II When then in the interval the selling rate

is defined as

Then the instantaneous inventory level at any time in are governed by the following differential equations

(24)

With boundary condition

(25)

With boundary condition

(26)

With boundary condition

The solution of the equation (21) is

(27)

The solution of the equation (22) is

(28)

(29)

From the continuity of , putting in the equation (24)

(30)

Using in the equation (24), the solution of the equation (21) is

(21A)

The maximum amount of demand backlogged per cycle is obtained by putting in the equation (26)

(31)

The maximum order quantity is

(32)

The ordering cost per cycle is

(33)

The holding cost per cycle is

(34)

The shortage cost per cycle is

(35)

The purchase cost per cycle is

(36)

The lost sales cost per cycle is

(37)

The sales revenue per cycle is

(38)

The total profit per unit time is

(39)

Now our objective is to determine the optimal value of for which the total profit is maximum.

The necessary condition for to be maximum is that

and solving these equations we find the optimum values of say for which profit is maximum and the sufficient condition is

(40)

(41)

(42)

(43)

(44)

4. Numerical Parameters

Let us consider the following parameters in the appropriate units and two different values of the ramp-type demand parameter

Numerical example 1- when then solving the equations we find the optimum value of and for different values of and

Table 1. Variation in total profit with respect to the change in deterioration parameter θ

Since the sign of total profit comes out to be negative and the value of so the total profit is minimum. As we increase the deterioration parameter then the total profit decreases.

Table 2. Variation in total profit with respect to the change in inflation parameter ρ

Since the sign of total profit comes out to be negative and the value of so the total profit is minimum. As we increase the inflation parameter then the total profit increases.

Numerical example 2- when then solving the equations we find the optimum value of and for different values of

Table 3. Variation in total profit with respect to the change in deterioration parameter θ

As we increase the deterioration parameter θ then the value of the total profit increases.

Table 4. Variation in total profit with respect to the change in inflation parameter ρ

As we increase the inflation parameter ρ then the value of the total profit increases.

5. Sensitivity Analysis

Thus from the tables 1, 2, 3 and 4 we see that the parameters are more sensitive in case of in comparison of because in the case I shortage interval is greater than the shortage interval in case II and so the total profit is minimum in case I and the total profit is maximum in case II.

6. Conclusions

In this paper we developed a probabilistic inventory model for deteriorating items with ramp-type demand rate under the effect of inflation. Shortages are allowed and partially backlogged. We studied the above developed model in two cases and . In case I from the tables 1 and 2 we see that as we increase the parameters and then the total profit is minimum. In case II from the tables 3 and 4 we see that as we increase the parameters and then the total profit is maximum. Thus when then the deterioration parameter and the inflation parameter become more sensitive in comparison of deterioration parameter and inflation parameter in case of because in the case I the shortage interval is greater than the shortage interval in case II. Further this model can be generalized by considering the fuzzy type demand rate.