. (14)

Since, there are n cycles in the planning horizon H, then the annual total relevant cost, which is a function of n, is given by

= n [Replenishment cost + Deterioration cost

+ Holding cost + Interest paid – Interest earn]

i.e.

(15)

Now, the total cost, when is given by

(16)

and, the total cost, when is given by

(17)

and, the total cost, when is given by

(18)

and, the total cost, when is given by

. (19)

According to the above arguments, we have

(20)

where, , and as expressed in Equations (16), (17), (18), and (19) respectively.

4. Algorithm

The following algorithm is developed to derive the optimal n, T, Q and values:

Step 1: Choose a discrete variable n first, where n is any integer equal or greater than 1.

Step 2: If, for different integer n values, obtain from Equation (16); If, for different integer n values, obtain from Equation (17); If, for different integer n values, obtain from Equation (18); If , for different integer n values, obtain from Equation (19).

Step 3: Repeat step 1 and 2 for all possible values of n with until the minimum is found from Equation (16) and let. For all possible values of n with until the minimum

is found from Equation (17) and let. For all possible values of n with until the minimum is found from Equation (18) and let. For all possible values of n with

until the minimum is found from Equation (19) and let. The, , , , , , , and values constitute the optimal solution.

Step 4: Select the optimal number of replenishment such that

(21)

Thus, optimal order quantity is obtained by putting into (3) and optimal cycle time is.

5. Numerical Example

To illustrate the results of the model developed in this study an example is given with the following data:

The demand rate, D = 960 unit/year, the replenishment cost, A = $60/order, the holding cost excluding interest charges, h = $1.5/unit/year, the per unit item cost, c = $ 3/unit, the deterioration rate, θ = 0.15, when account is settled at M, the interest charged per $ in stocks per year by the supplier/retailer, I_{c} = $0.18/$/year, the interest earned, I_{e} = $0.16/$/year, when account is settled at N, the interest charged per $ in stocks per year by the supplier/retailer, I_{w} = $0.21/$/year, and the planning horizon, H = 5 year. The delay in payment in settling account, M = 0.083 year, and N = 0.14 year, assuming 360 days per year. Using the algorithm, we have the computational results shown in Table 1.

The graphical representation of total cost when M ≤ T, M > T, M < N ≤ T and N > T > M are given in Figures 2(a)-(d) respectively which are given below.

In the numerical example, we see that the case I is optimal option in credit policy when the account is settled at “M”. The minimum total present value of costs is obtained when the number of replenishment, n, is 20. With 20 replenishments, the optimal (minimum) cycle time is 0.250 year, the optimal (minimum) order quantity, Q = 244.500 units and the optimal (minimum) total present value of costs, = $2226.5765. Also, the case III is optimal option in credit policy when the account is settled at “N”. The minimum total present value of costs is obtained when the number of replenishment, n, is 20.

Table 1. Computational analysis for optimal total cost.

(a)(b)(c)(d)

Figure 2. Plot of “n” against TVC (n).

With 20 replenishments, the optimal (minimum) cycle time is 0.250 year, the optimal (minimum) order quantity, Q = 244.500 units and the optimal (minimum) total present value of costs, = $2237.0309.

In the total optimal cost, we see that, there is miner difference between the costs when the account is settled at “M” and at “N”. Thus, the model is more realistic to the business environment and gives more freedom to the suppliers/retailers to run business smoothly. By this policy, they can attract the customers and increase their sales. Therefore, the policy involves inventory, financing and marketing issue. So, we investigate that this model is very important and valuable to the enterprises.

6. Sensitivity Analysis

Taking all the parameters as in the above numerical example, the variation of the optimal solution for different values of M is given in Table 2, and variation of the optimal solution for different values of θ is given in Table 3.

The data obtained clearly shows those individual optimal solutions different from each other for different values of trade credit M, and the deterioration rate θ. All the observations can be summed up as follows:

1) In the total four cases, , ,

Table 2. Sensitivity analysis on M.

Table 3. Sensitivity analysis on θ.

and, an increase in the trade credit M, the total costs are in decreasing trend.

2) Observed in Table 3, when deterioration rate (θ) increases then in the total four cases, , , and, the total costs are in increasing trend.

7. Conclusion and Future Research

This study develops an inventory model for deteriorating items over a finite planning horizon, when the suppliers/ retailers provide two levels of permissible delay in payments in one replenishment cycle. The model considers the effects of permissible delay in payments at both stages. In addition, an optimal solution procedure is obtained to find the optimal number of replenishment, cycle time, and order quantity to minimize the total present value of cost. The proposed model can be extended in several ways, such as for time dependent demand and deterioration. The inflation rate may also be introduced. Finally, further research can be done with stochastic market demand and partial backlogging.

8. Acknowledgements

The first author is grateful to the CSIR, New Delhi for providing the financial assistance as JRF.

REFERENCES

- K. J. Chung, “A Theorem on the Determination of Economic Order Quantity under Conditions of Permissible Delay in Payments,” Computers & Operations Research, Vol. 25, 1998, pp. 49-52.
- Y. F. Huang, “Optimal Retailer’s Ordering Policies in the EOQ Model under Trade Credit Financing,” Journal of Operational Research Society, Vol. 54, No. 9, 2003, pp. 1011-1015. doi:10.1057/palgrave.jors.2601588
- K. J. Chung and J. J. Liao, “Lot-Sizing Decisions under Trade Credit Depending on the Ordering Quantity,” Computers & Operations Research, Vol. 31, No. 6, 2004, pp. 909-928. doi:10.1016/S0305-0548(03)00043-1
- A. Jamal, B. Sarker and S. Wang, “An Ordering Policy for Deteriorating Items with Allowable Shortage and Permissible Delay in Payment,” Journal of Operational Research Society, Vol. 48, No. 8, 1997, pp. 826-833.
- H. J. Chang and C. Y. Dye, “An Inventory Model for Deteriorating Items with Partial Backlogging and Permissible Delay in Payments,” Journal of Systems Science, Vol. 32, No. 3, 2001, pp. 345-352. doi:10.1080/002077201300029700
- K. N. Huang and J. J. Liao, “A Simple Method to Locate the Optimal Solution for Exponentially Deteriorating Items under Trade Credit Financing,” Computers and Mathematics with Applications, Vol. 56, No. 4, 2008, pp. 965-977. doi:10.1016/j.camwa.2007.08.049
- V. B. Kreng and S. J. Tan, “The Optimal Replenishment Decisions under Two Levels of Trade Credit Policy Depending on the Order Quantity,” Expert Systems with Applications, Vol. 37, No. 7, 2010, pp. 5514-5522. doi:10.1016/j.eswa.2009.12.014
- L. Y. Ouyang, K. S. Wu and C. T. Yang, “A Study on an Inventory Model for Non-Instantaneous Deteriorating Items with Permissible Delay in Payments,” Computers and Industrial Engineering, Vol. 51, No. 4, 2006, pp. 637-651. doi:10.1016/j.cie.2006.07.012
- K. L. Hou and L. C. Lin, “A Cash Flow Oriented EOQ Model with Deteriorating Items under Permissible Delay in Payments,” Journal of Applied Sciences, Vol. 9, No. 9, 2009, pp. 1791-1794. doi:10.3923/jas.2009.1791.1794