1. Introduction

Inventory model is an important part of operations research, which may be used in variety of problems. To make it applicable in real life situation researchers are engaged in modifying existing models on different parameters under various circumstances. Inventories play a major role in the united stages economy and have been in excess of 22% of the nation’s gross national product over the past few decades. As millions of dollars are tied up in inventories, proper management of these inventories can lead the organisation or company to become profitable in the entire globe. A major concern of inventory management is to know when and how much to order or manufacture so that the total cost per unit time is minimized. The total cost consists of carrying, shortage, replenishment or setup cost and the purchase or production cost. In the development of an EOQ system, we usually omit the case of inflation of money, promotional activities and deterioration of the items like fruit, milk, drug, chemicals, gasoline etc. But in the real competitive world they exist and are quite flexible in nature. On the other hand to attract customer to order more quantities, usually suppliers offer a certain credit period without interest during the permissible delay time period. Inventory management plays a significant role for production system in business since it can help companies reach the goal of ensuring prompt delivery, avoiding shortages, helping sales at competitive prices and so forth for achieving competitive advantage. However, excessive simplification of assumptions results in mathematical models that do not represent the inventory situation to be analyzed.

The classical analysis builds a model of an inventory system and calculates the EOQ which minimize the costs satisfying minimization criterion. One of the unrealistic assumption is that items stocked preserve their physical characteristics during their stay in inventory for long run. Items in stock are subject to many possible risks, e.g. damage, spoilage, dryness, vaporization etc., those results decrease of usefulness of the original one and a cost is incurred to account for such risks of the product.

The problem of deteriorating inventory has received considerable attention in recent years. This is a realistic trend since most products such as medicine, diary products and chemicals starts to deteriorate once they are produced.

Most researches in inventory do not consider inflation with promotional activities. This is unrealistic, since the resource of an enterprise depends very much on when it is used and this is highly correlated to the return of investment. Therefore, taking into account the inflation of money should be critical especially when investment and forecasting are considered.

Buzacott (1975) ^[1] explained inventory model to determine economic order quantities with inflation. Datta and Pal (1991) ^[2] verified the effect of inflation and time value of money on an inventory model with linear time dependant demand rate and shortages. Haneveld and Teunter (1998) ^[3] discussed the inventory model where they studied the effects of discounting and demand rate variability on the EOQ. Holland (1995) ^[4] derived an inventory model where inflation and uncertainty tests for temporal ordering are allowed. Horowitz (2000) ^[5] presented an inventory model for obtaining EOQ under uncertain inflation of the economy. Liao et al. (2000) ^[6] derived an inventory model with deteriorating items under inflation when a delay in payment is permissible. Mishra (1979) ^[7] focused an optimal inventory management policy under inflation. Moon and Lee (2000) ^[8] verified the effect of inflation and time value of money on an EOQ order quantity model with random product life cycle. Pattnaik, M. (2012) ^[9] and Pattnaik (2014) ^[10] derived different types of deterministic inventory models for deteriorating items in finite horizon. Shah (1993) ^[11] discussed a lot size EOQ model for exponentially decaying inventory when delay in payments is permissible. Skouri and Papachristors (2002) ^[12] developed deterministic inventory lot-size models under inflation with shortages and deterioration for the fluctuating demand by Yang et al. Their approach was performed in the framework of the total inventory cost under inflation criterion.

The objective of this paper is to minimize the inventory cost so as to determine the optimal order quantity and promotional effort factor under inflation where shortages are allowed. The remainder of paper organized in section 2 is assumptions and notations for development of the model. The mathematical model is developed in section 3. Optimization is given in section 4. Numerical example is presented to illustrate the development of model in section 5. The sensitivity analysis carried out in section 6 to observe the changes in optimal solution. Finally section 7 deal with conclusion.

2. Assumption and Notation

The mathematical model in this paper is developed with the following assumptions:

1. The replenishment is instantaneous.

2. The lead time is zero.

3. Backlogging is allowed within a small interval of time.

4. The total demand for the whole planning period is unknown and constant.

5. Discounts are not allowed and a permissible delay period will not exceed the length of a cycle.

6. During the fixed credit period M, a deposit is made from the unit cost of generated sales revenue into an interest-bearing account. The delay expenses of the system can be overcome from the difference between retail price and unit cost. At the end of period M, the account is settled, and interest changes are payable on the account in stock.

Notations

The following notations are used

T: Time period of each cycle

t₁: Time from where shortage begins

M: Permissible delay period for settling accounts

H: Length of the planning horizon (H=mT, m>1 is a positive integer)

I₁(t): Inventory level at any time

I₂(t): Inventory level at any time

Q: Order size per cycle

D: Total demand per year

S: Shortage quantity per cycle

K: Rate of inflation ($/unit time)

Θ: Deterioration rate (<1) per unit time

H: Holding cost per unit/year

S₀: Unit shortage cost per unit time

C(t): Unit purchasing price for an item bought at time t, where c(t)=c0(1+kt), 0≤t≤(m-1)t, c0 is the unit price of the item at time zero

A(t): Ordering cost for an order placed at time t, i.e. A(t)=A₀(1+kt), 0≤t≤(m-1)t, A₀ is unit ordering cost at time zero

i_e: Annual interest that can be earned per unit

i_c: Annual interest charges payable per until where i_c>i_e.

Ρ: The promotional effort factor per cycle.

PE(ρ): The promotional effort cost, , where, K₁>0 and α₁ is a constant.

3. Mathematical Model

The total period H is divided into m equal sub-intervals of constant length T such that H-mT. For a subinterval (0,T) the inventory Q is received at time t=0. Due to demand and deterioration, the inventory level reaches zero level at time t=t₁ and shortages start to occur. The shortage continue upto t=T and accumulates to S quantity. If I₁(t) and I₂(t) denotes the inventory lee at any time t where and respectively. Figure 1 represents the inventory level for delay in payment. Then the differential equation

(1)

(2)

With

(3)

Solving equation(1) and (2)

Equation (1) is linear differential equation.

Now Integrating factor

Multiplying both side of equation (1) by Integrating factor we get

(4)

By using initial condition I₁ (t₁) = 0

From equation (4)

So

(5)

From equation (2)

From initial condition I₂ (t₁)= 0 when t = t₁

O = –Dt₁ + C₁

Hence the solution is,

(6)

Shown in appendix A

(7)

and

(8)

(as and higher order)

Case-1:M<t₁

As A(t) = replenishment cost at time t

C(t) = Purchasing cost at time t

C_r = Replenishment cost in (O,H)

Given that

(9)

The purchasing cost C_p in (O,H) is

Given C(t) = C₀ (1+Kt) where 0<t < (m–1) T

(10)

The holding cost in (O,H)

(11)

The shortage cost in (O,H) is

But given C(t) = C₀ (1+kt)

(12)

The interest charged for the inventory not being after the due date M in (0,H) is

(13)

Detail calculation of and are given in appendix A.

(14)

From equation (9) – (14) the total system cost over (0,H) is given by

Where

Case –II (M = t₁)

Thus using equation 9-12 and 16 we have the total system cost over (0,H)

Where

Case-III

No interest is charged in (0,H) because the supplier can be paid in full at the time of permissible delay and hence the interest earned in (0,H) is

Using and and

The total system cost over (0,H)

+

=

+

Where

4. Solution Procedure

Choosing two initial values of m^* say m and (m–1) in Z(m). Compute T^*= Hm^* and then optimize Z(m) and Z(m–1) with the help of LINGO software to find the parameters Q^*, S^*, t₁^* in each case.

Step-1:

If Z^* (m) then set m^* = k and stop.

5. Numerical Example

Here we have consider following numerical example.

Let A₀ = $ 7.5 ,C₀= $2.5, M = 0.3yr (case 1 only)

I_C = $0.11, i_e = $0.09, D = 500 units per year.

h=$0.18, H = 1 year, K = 0.5, Q = 0.004

S₀ = $0.5, k₁ = 2.00,

For the given data the total net value profit for the planning horizon is Rs.74703.13, the number of replenishment during planning horizon N, is 26, per unit selling price of the items is Rs. 18.52, time with positive inventory T₁, is 0.2649512, replenishment cycle T is 0.3856304, selling price dependent declining quadratic demand d(s) is 570.3798 and the replenishment size Q is 220.0833. The total number of order is therefore N+1=27. All the decision parameters are compared with the other model related to the declined demand d(S) which is also related linearly to the selling price. It is observed that demand rate d(s), order quantity Q, time with positive inventory and replenishment cycle T are more than the compared model. But only the number of replenishment during planning horizon, N, selling price S, unit selling prices and the total present value profit are less from the compared model.

6. Sensitivity Analysis

It is interesting to investigate the influence of major parameters

by taking six cases such that are represented in appendix- B.

Case-:

●   are insensitive to the parameter

●   are insensitive to the parameter but TC is moderately sensitive to

●   are insensitive to is moderately sensitive to but TC is highly sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter to but TC is moderately sensitive to

●   are insensitive to D. Q and TC are moderately sensitive to D.

●   are insensitive to h.

●   are insensitive to parameter H.

●   are insensitive to parameter K.

●   are insensitive to parameter

●   are insensitive to parameter

Case-:

●   are insensitive to the parameter

●  S and Q are moderately sensitive to the parameter

.

●   are insensitive to the parameter

●   are insensitive to TC is moderately sensitive to but S and Q are highly sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter to but S and Q are highly sensitive to the parameter TC is moderately sensitive to

●   are insensitive to D. TC are moderately sensitive to D.

●   are insensitive to h. S and Q are highly sensitive to h and TC is moderately sensitive to h.

●   are insensitive to parameter H and S, Q and TC are moderately sensitive to H.

●   are insensitive to parameter K but S, Q and TC are moderately sensitive to the parameter K.

●   are insensitive to parameter

●   are insensitive to parameter but S is highly sensitive to Q is moderately sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter

Case-

●   are insensitive to the parameter

●  S and TC are moderately sensitive to are insensitive to the parameter

●   are insensitive to S, Q and TC are moderately sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter to but S is highly sensitive to the parameter Q and TC is moderately sensitive to

●   are insensitive to D. Q and TC are moderately sensitive to D.

●   are insensitive to h. S, Q and T are moderately sensitive to h.

●   are insensitive to parameter H and S, Q and TC are moderately sensitive to H.

●   are insensitive to parameter K but S, Q and TC are moderately sensitive to the parameter K.

●   are insensitive to parameter

●   are insensitive to parameter but S and Q is moderately sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter

Case-

●   are insensitive to the parameter

●   are insensitive to the parameter

●   are insensitive to Q and TC are moderately sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter to but TC is moderately sensitive to

●   are insensitive to D. Q, PE and TC are moderately sensitive to D.

●   are insensitive to h. TC is moderately sensitive to h.

●   are insensitive to parameter H and S, Q and TC are moderately sensitive to H.

●   are insensitive to parameter K but S, Q and TC are moderately sensitive to the parameter K.

●   are insensitive to parameter

●   are insensitive to parameter but S and Q is moderately sensitive to ...

●   are insensitive to the parameter

●   are insensitive to the parameter

Case-B2= Case-B3

●   are insensitive to the parameter

●  S and TC are moderately sensitive to are insensitive to the parameter

●   are insensitive to S, Q and TC are moderately sensitive to

●   are insensitive to the parameter

●   are insensitive to the parameter to but S is highly sensitive to the parameter Q and TC is moderately sensitive to

●   are insensitive to D. Q and TC are moderately sensitive to D.

●   are insensitive to h. S, Q and T are moderately sensitive to h.

●   are insensitive to parameter H and S, Q and TC are moderately sensitive to H.

●   are insensitive to parameter K but S, Q and TC are moderately sensitive to the parameter K.

●   are insensitive to parameter

●   are insensitive to parameter but S and Q is moderately sensitive to

●   are insensitive to the parameter

●  are insensitive to the parameter

7. Conclusion

In this paper, an EPQ model is introduced which investigates the optimal replenishment quantity. Unit selling price, replenishment cycle, time with positive invention the total value net profit with finite planning horizon for deteriorating items. The model considers the impact of price dependant quadratic demand, and shortages and varying rate of deterioration. The model can be used for electronics and other luxury products which are more likely to have the above characteristics. This paper provides a useful property for finding the optimal net present value profit with finite planning horizon for deteriorating items. A new mathematical model with decline quadratic demand is developed and compared to the other EPQ model with decline linear demand and numerically. The economic replenishment quantity Q^* and net present value profit for the present model were found to be more than that of the compared model respectively. Hence the utilization of selling price dependent declined quadratic demand makes the scope of application broader. Lingo 13.0 version software is used to derive the optimal number of replenishment and unit price to maximize the total present value net profit. Further, a numerical example is presented to illustrate the theoretical results, and some observations are obtained from sensitivity analysis with respect to the major parameters controlling the market demand through the manipulation of selling price is an important strategy for increasing profit. This can be achieved by using the joint optimal replenishment and pricing strategy developed in this study. In the future study, it is hoped further incorporate the proposed model into several situations such as stochastic market demand, fuzzy decision parameters and partial back logging.

References

[1]	Buzacott, J.A. (1975). Economic order quantities with inflation. Operations Research Quarterly, 26: 553-558.
	In article		CrossRef
[2]	Datta, T.K. and Pal, A.K. (1991). Effect of inflation and time value of money on an inventory model with linear time dependant demand rate and shortages. European Journal Operational Research, 52: 326-333.
	In article		CrossRef
[3]	Haneveld, W.K. and Teunter, R.H. (1998). Effects of discounting and demand rate variability on the EOQ. International Journal of Production Economics, 54: 173-192.
	In article		CrossRef
[4]	Holland, A.S. (1995). Inflation and uncertainty tests for temporal ordering. Journal of Money Credit Bank, 27: 827-837.
	In article		CrossRef
[5]	Horowitz, I. (2000). EOQ and inflation uncertainty. International Journal Production Economics, 65: 217-224.
	In article		CrossRef
[6]	Liao, H.C., Tsai C.H. and Su, C.T. (2000). An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal Production Economics, 63: 207-214.
	In article		CrossRef
[7]	Mishra, R.B. (1979). A note an optimal inventory management under inflation, Naval Research Logistic Quarterly, 26: 161-165.
	In article		CrossRef
[8]	Moon, I. and Lee, S. (2000). The effect of inflation and time value of money on an EOQ order quantity model with random product life cycle, European Journal of Operational Research, 125: 588-601.
	In article		CrossRef
[9]	Pattnaik, M. (2012). Models of Inventory Control, Lambart Academic Publishing, Germany.
	In article
[10]	Pattnaik, M. (2014). Inventory Models: A Management Perspective, Lambart Academic Publishing, Germany.
	In article
[11]	Shah, N.H. (1993). A lot size model for exponentially decaying inventory when delay in payments is permissible, Cahiers du CERO, 35: 115-123.
	In article
[12]	Skouri, K. and Papachristors, S. (2002). Note on deterministic inventory lot-size models under inflation with shortages and deterioration for the fluctuating demand by Yang et al. Naval Research Logistics, 49: 527-529.
	In article		CrossRef

Appendix -A

From equation A₁ we have

Since

as

The holding cost (O, H)

Now

Hence

But we know that

Neglect higher part of

Hence

Hence

Hence

Appendix -B