Abstract: In this study, an order-level inventory model is constructed for deteriorating items with instantaneous replenishment, exponential decay rate and a time varying linear demand without shortages under per-missible delay in payments. It is assumed that a constant fraction of the on-hand inventory deteriorates per unit of time. The exact formulae of the optimal average cost and the lot size are derived without carrying out any approximation over the deterioration rate. Different decision making situations illustrated with the help of numerical examples. Sensitivity analysis of the optimal solution with re-spect to changes in the parameter values is carried out.

INTRODUCTION

In formulating inventory models, two factors of the problem have been of growing interest to the researchers, one being the deterioration of items and the other being the variation in the demand rate.

Demand is the major factor in the inventory management. Therefore, decisions of inventory are to be made because of the present and future demands. As demand plays a key role in modeling of deteriorating inventory, researchers have recognized and studied the variations (or their combinations) of demand from the viewpoint of real life situations. Demand may be constant, time-varying, stock-dependent and price-dependent, etc. The constant demand is valid only when the phase of the product life cycle is matured and also for finite periods of time. Wagner and Whithin (1958) discussed the discrete case of the dynamic version of EOQ. Covert and Philip (1973), Misra (1975), Dave (1979) and Sarma (1987), etc., established inventory models with constant demand rate. The classical no-shortage inventory policy for linear trend in demand was discussed by Donaldson (1977). Ritchie (1980, 1984) discussed the solution of the EOQ formula for a linear increasing time-dependent demand which was obtained by Donaldson (1984). EOQ models for deteriorating items with trended demand were considered by Bahari-Kasani (1989), Goswami and Chaudhuri (1991, 1992), Xu and Wang (1990), Chung and Ting (1993, 1994), Kim (1995), Jalan et al. (1996), Jalan and Chaudhuri (1999) and Lin et al. (2000), etc. In this extent, Begum et al. (2009) developed an inventory model with exponential demand rate, finite production rate and shortages. Silver and Meal (1969) developed an approximate solution procedure for the general case of time-varying demand. Generally, this type of demand exists for some particular goods. Many research articles by Silver (1979), Henery (1979), McDonald (1979), Dave and Patel (1981), Sachan (1984), Deb and Chaudhuri (1986), Murdeshwar (1988) and Hargia (1993), etc., analyzed linear timevarying demand. Later, Ghosh and Chaudhuri (2004, 2006), Khanra and Chaudhuri (2003) and Begum et al. (2010), etc., established their models with quadratic time-varying demand. Teng and Chang (2005) established an economic production quantity models for deteriorating items with demand depend on price and stock. The deterioration rate of inventory in stock during the storage period constitutes an important factor which has attracted the attention of researchers. In inventory problems, deterioration is defined as damage, decay, spoilage, evaporation, obsolescence and loss of utility or loss of marginal value of goods that results in decrease the usefulness of the original one. Whitin (1957) is the first reseacher who studied an inventory model for fashion goods deteriorating at the end of a prescribed storage period. An exponentially decaying inventory was developed by Ghare and Schrader (1963). Emmons (1968) established a replenishment model for radioactive nuclide generators.

The assumption of the constant deterioration rate was relaxed by Covert and Philip (1973) who used a two parameter Weibull distribution to represent the distribution of time to deterioration. This model was further generalized by Philip (1974) by taking three parameter Weibull distribution deterioration. Shah and Jaiswal (1977) established an order-level-inventory model for perishable items with a constant rate of deterioration.

Recently, Begum et al. (2010) has discussed an EOQ model for the deteriorating items with two parameter Weibull distribution deterioration.

The problem of determining the Economic Order Quantity (EOQ) under the condition of a permissible delay in payment has drawn the attention of researchers in recent times. It is assumed that the supplier (wholesaler) allows a delay of a fixed period for settling the amount owed to him/her. There is no interest charged on the outstanding amount if it is paid within the permissible delay period.

Beyond this period, interest is charged. During this fixed period of permissible delay in payments, the customer (a retailer) can sell the items, invest the revenues in an interest-earning account and earn interest instead of paying off the over-draft which is necessary if the supplier requires settlement of the account immediately after replenishment. The customer finds it economically beneficial to delay the settlement to the least moment of the permissible period of delay. This problem was first studied by Goyal (1985) for a non-deteriorating items having a constant demand rate. Chand and Ward (1987) commented in a brief note on some of the assumptions made by Goyal (1985) in analyzing the cost of funds tied up in inventory. The effects of deterioration of goods in stock on the cost and price components cannot be ignored in practice. The model of Goyal (1985) was extended by Aggarwal and Jaggi (1995) to the case of a deteriorating item. Hwang and Shinn (1997) discussed lot sizing policy for an exponentially deteriorating products under the condition of permissible delay in payments when the demand rate would depend on retail price.

In the present study, the researchers assume that the time dependence of demand is linear. Deterioration rate is assumed to be exponential distribution. The rate of replenishment is infinite and shortages are not allowed. The results presented in this study extend and improve the corresponding results of Aggarwal and Jaggi (1995) by taking into account a time-dependent demand rate. The solution procedure involving different decission making situations is illustraded with the help of numerical examples. Analysis is carried out to study sensitivity of the optimal solution to changes in the values of the different parameters involved in the system.

NOTATION AND ASSUMPTIONS

The mathematical model is developed on the basis of the assumptions and notations.

Notation:

Assumptions:

Where, θ is called the deterioration rate; a constant fraction θ, assumed to be small of the on-hand inventory gets deteriorated per unit time during the cycle time.

MATHEMATICAL FORMULATION

(1)

Where:

(2)

And:

(3)

Where, a>0, b>0. The solution of Eq. 1 using Eq. 2 and 3 is:

(4)

The initial order quantity at t = 0 is:

(5)

The total demand during one cycle is:

Number of deteriorated units:

(6)

The cost of stock holding for one cycle is:

Hence, the holding cost per unit time is:

(7)

Case 1
Let T>M: Since, the interest is payable during time (T-M), the interest payable in one cycle is:

Hence, interest payable per unit time is:

(8)

Interest earned per unit time is:

(9)

Total variable

Hence, total variable cost per unit time in this case is given by:

(10)

The researchers have now to minimize C₁(T) for a given value of M. The necessary and sufficient conditions to minimize C₁ (T) for a given value of M are respectively:

(11)

And:

(12)

After simplification:

yields the following nonlinear equation in T:

(13)

By solving Eq. 13 for T, we obtain the optimal cycle length T = T₁* provided it satisfies Eq. 12. The EOQ q₀* for this case is given by:

The minimum annual variable cost C₁(T₁*) is then obtain from Eq. 10 for T = T₁*.

Case 2
T<M: In this case, the customer earns interest on the sales revenue up to the permissible delay period and no interest is payable during this period for the items kept in stock. Interest earned up to T is:

and interest earned during (M–T) i.e., up to the permissible delay period is:

Hence, the total interest earned during the cycle is:

(14)

Hence, the total variable cost per unit time is:

(15)

The researchers have now to minimize C₂(T) as before for a given value of M. After simplification:

yields the result:

(16)

The optimal cycle length T = T₂* which minimizes C₂(T) is obtained by solving Eq. 16 for T by using the Newton-Raphson method, provided:

The EOQ in this case is given by:

and the minimum annual variable cost C₂(T₂* ) is obtained from Eq. 15 for T = T₂*.

Case 3
T = M: For T = M, both the cost functions C₁(T) and C₂(T) become identical and it is denoted by C(M), say and it is obtained on substituting T = M either in Eq. 10 or in Eq. 15. Thus:

(17)

The EOQ is:

Now in order to obtain the economic operating policy, the following steps are to be followed:

NUMERICAL EXAMPLES

The numerical examples given, covers all the three cases that arise in this model:

Example 1 (Cases 1 and 2): Let a = 1000 units year^-1, b = 150 units year^-1, I_p = 0.15 year^-1, I_e = 0.13 per year, s = Rs. 200 per order, h_p = Rs. 0.12 year^-1, p = Rs. 20 per unit, M = 0.25 year, θ = 0.20.

Solving Eq. 13, we have T₁* = 0.284 and the minimum average cost is C₁(T₁*) = 1283.53. Solving Eq. 16, the researchers have T₂* = 0.206 and the minimum average cost is C₂(T₂*) = 1263.53. Here T₁*>M and T₂*<M both hold and this implies that both the cases 1 and 2 hold. Now C₂(T₂*)<C₁(T₁*). Hence, the minimum average cost in this case is C₂(T₂*) = Rs. 1263.53 where the optimal cycle length is T₂* = 0.206 year <M. The economic order quantity is given by q₀* (T₂*) = 213.82 units.

Example 2 (Case 1): Let a = 1000 units year^-1, b = 150 units year^-1, I_p = 0.15 year^-1, I_e = 0.13 year^-1, s = Rs. 200 per order, h_p = Rs. 0.12 year^-1, p = Rs. 20 per unit, M = 0.25 year, θ = 0.01.

Solving Eq. 13 for T, the researchers get T₁* = 0.432 and the minimum average cost is C₁(T₁*) = 585.31. Again, solving Eq. 16, the researchers have T₂* = 0.274 and the corresponding minimum average cost is C₂(T₂*) = 793.94.

Here T₂*>M which contradicts case 2. In this case T₁>M which is case 1. Therefore, the minimum average cost in this case is C₁(T₁*) = Rs. 585.31, the EOQ is q₀* (T₁*) = 447.23 units and the optimal cycle length is T_ 1 = 0.432 year >M.

Example 3 (Case 2): Let a = 1000 units year^-1, b = 150 units year^-1, I_p = 0.15 year^-1, I_e = 0.13 year^-1, s = Rs. 200 per order, h_p = Rs. 0.12 year^-1, p = Rs. 40 per unit, M = 0.25 year, θ = 0.20.

Solving Eq. 13 for T, the researchers obtain the optimal value T₁* = 0.232 and the optimal cost C₁ (T₁*) = 1792.29. Here, T₁*<M which contradicts case 1. Again, solving Eq. 16, the researchers have T₂* = 0.147 and the minimum average cost is C₂ (T₂*) = 1395.29. In this case, T₂*<M which is case 2. Therefore, the minimum average cost in this case is C₂ (T₂*) = Rs. 1395.29, the EOQ is q₀* (T₂*) = 150.81 units and the optimal cycle length is T₂* = 0.147 year <M.

Example 4 (Case 3): Let a = 1300 units year^-1, b = 100 units year^-1, I_p = 0.5 year^-1, I_e = 0.01 year^-1, s = Rs. 97 per order, h_p = Rs. 0.12 year^-1, p = Rs. 40 per unit, M = 0.09 year, θ = 0.3.

In this case T₁* = T₂* = 0.09 = M which is case 3. The optimal cost in this case is C (M) = Rs. 2050.56 and the EOQ is q₀* = 119.01 units for M = 0.09 year.

Table 1:	Sensitivity of the optimal solution to changes in parameter values

Table 2:	Results for p = Rs. 20 per unit

SENSITIVITY ANALYSIS

The researchers now study sensitivity of the solution of the problem to changes in the values of the parameters of the systems. Example 3 is used for this purpose and the results are shown in Table 1. It is found that the solution is not sensitive to changes in the values of the parameters b and Ip. However, it is sensitive to changes in the values of the parameters a, h_p, I_e, p, s, θ and M.

In Table 2, we present the optimal solutions, as M and θ vary, for a less expensive item (p = Rs. 20) when the parameter values are a = 1000, b = 150, h_p = 0.12, I_p = 0.15, I_e = 0.13 and s = 200 in appropriate units. We observe the the following characteristics of the solution:

Table 3:	Results for p = Rs. 40 per unit

A similar analysis is made in Table 3 for a more expensive (p = Rs. 40) item. The same type of results is observed in this case also.

Table 4:	Results for p = Rs. 200 per unit

In Table 4, it is observed that the cycle length, the order quantity and the average system cost all undergo considerable changes for a very expensive (p = Rs. 200) item.

CONCLUSION

The present model seeks an extension of Aggarwal and Jaggi (1995) work by taking a time-dependent demand rate into consideration. This consideration makes the model more realistic. A linear, time-dependent demand rate implies a steady increase in the demand of the product. This type of demand pattern is observed in the market in the case of many products. Several numerical illustrations are given to explain the solution procedure of the model. Sensitivity analysis is also carried out. Comparison of this model with that of Aggarwal and Jaggi (1995) shows that there are slight changes in the optimal solutions when a time-dependent demand rate is taken into consideration. As a result of the demand rate varying linearly with time, both the cycle length and the order quantity decreases whereas the average system cost increases in comparison to the results of Aggarwal and Jaggi (1995).