# Integrated inventory system with the effect of inflation and credit period.

Introduction

Many delivery policies have been proposed in literature for this problem. Clark and Scarf (1960) presented the concept of serial multi-echelon structures to determine the optimal policy. Ha and Kim (1997) used a graphical method to analyze the integrated vendor-buyer inventory status to derive an optimal solution. Yang and Wee (2000) developed an integrated economic ordering policy of deteriorating items for a vendor and a buyer. Other researches related to this area such as Lee and Wu (2006), Chen and Kang (2007), Singh (2008).

Buzacott (1975) who discussed EOQ model with inflation subject to different types of pricing policies. Bose et al. (1995) presented a paper on deteriorating items with linear time dependent rate and shortages under inflation and time discounting. Chang (2004) proposes an inventory model for deteriorating items under inflation under a situation in which the supplier provides the purchaser a permissible delay of payments if the purchaser orders a large quantity. Yang (2006) considered partial backlogging inventory models for deteriorating items under inflation. Lo et.al. (2007) developed an integrated production and inventory model from the perspectives of both the manufacturer and the retailer assuming inflation. Hwang and Shinn (1997) studied effects of permissible delay in payments on retailer's pricing and lot sizing policy for exponentially deteriorating products. Wang et al. (2000) analyzed supply chain models for perishable products under inflation and permissible delay in payment. Chung and Huang (2003) studied the optimal cycle time for EPQ inventory model under permissible delay in payments. Huang (2005) considered the optimal inventory policies under permissible delay in payments depending on the ordering quantity. Song and Cai (2006) has been taken on optimal payment time for a retailer under permitted delay of payment by the wholesaler. Liao (2007) assumed on an EPQ model for deteriorating items under permissible delay in payments..

In the present paper we combine all the above mentioned factors into a single problem. We shall undertake to explore a two echelon supply chain, comprising of a vendor, a buyer. The demand rate is increasing exponentially due to inflation. The whole environment of business dealings has been assumed to be progressive credit period, which conforms to the practical market situation. The whole combination is very unique and very much practical. The setup has been explored numerically as well; an optimal solution has been reached. The final outcome shows that the model is not only economically feasible, but stable also.

Assumptions and Notations

The following assumptions are used to develop aforesaid model:

The demand rate is exponentially increasing and is represented by D(t) = [[lambda].sub.0][e.sup.[alpha]t], where 0[less than or equal to] [alpha] [less than or equal to] 1 is a constant inflation rate and [[lambda].sub.0] is the initial demand rate.

1. Shortages are allowed.

2. If the retailer pays by M, then the supplier does not charge to the retailer. If the retailer pays after M and before N (N > M), he can keep the difference in the unit sale price and unit purchase price in an interest bearing account at the rate of [I.sub.e]/unit/year. During [M, N], the supplier charges the retailer an interest rate of [Ic.sub.1]/unit/year on unpaid balance. If the retailer pays after N, then supplier charges the retailer an interest rate of [Ic.sub.2]/unit/year ([Ic.sub.1] > [Ic.sub.2]) on unpaid balance.

The notations are as follows:

1. [T.sub.1] = the length of production time in each production cycle

2. [T.sub.2] = the length of non production time in each production cycle

3. [I.sub.v1]([t.sub.1]) = inventory level for vendor when [t.sub.1] is between 0 and [T.sub.1]

4. [I.sub.v2]([t.sub.2]) = inventory level for vendor when [t.sub.2] is between 0 and [T.sub.2]

5. [I.sub.b](t) = inventory level for buyer when t is between 0 and T/n

6. n = delivery times per period T for buyer

7. P = the selling price/unit.

8. C = the unit purchase cost, with C < P.

9. M = the first offered credit period in settling the account without any charges.

10. N = the second permissible credit period in settling the account with interest charge Ic2 on unpaid balance and N > M.

11. Ic1 = the interest charged per \$ in stock per year by the supplier when retailer pays during [M, N].

12. Ic2 = the interest charged per \$ in stock per year by the supplier when retailer pays during [N, T]. (Ic1 > Ic2)

13. Ie = the interest earned/\$/year.

14. T = the replenishment cycle.

15. IE = the interest earned/time unit.

16. IC = the interest charged/time unit.

17. [C.sub.vs] = the setup cost for each production cycle for vendor.

18. [C.sub.bs] = the setup cost per order for buyer.

19. [C.sub.vh] = holding cost per unit time for vendor.

20. [C.sub.bh] = holding cost per unit time for buyer.

21. [C.sub.v] = the unit production cost for vendor.

22. [C.sub.b] = the unit price for buyer.

23. [S.sub.b] = shortage cost per unit time for buyer.

24. r = the discount rate (r > [alpha])

25. VC = the cost of vendor per unit time.

26. BC = the cost of buyer per unit time.

27. TC([T.sub.2]) = total cost of an inventory system/time unit.

The actual vendor's average inventory level in the integrated two-echelon inventory model is difference between the vendor's total average inventory level and the buyer's average inventory level.

Since the inventory level is depleted due to a constant deterioration rate of the onhand stock, the buyer's inventory level is represented by the following differential equation:

[I.sup.'.sub.b](t) + [theta][I.sub.b](t) = -[[lambda].sub.0][e.sup.[alpha]t], 0 [less than or equal to] t [less than or equal to] [t.sub.1] (1)

[I.sup.'.sub.b](t) = -[[lambda].sub.0][e.sup.[alpha]t], [t.sub.1] [less than or equal to] t [less than or equal to] T (2)

The vendor's total inventory system consisting of production period and non-production period can be described as follows:

[I.sub.'.sub.v1](t) + [theta][I.sub.v1](t) = -(K - 1)[[lambda].sub.0][e.sup.[alpha]t], 0 [less than or equal to] t [less than or equal to] [T.sub.1] (3)

[I.sub.'.sub.v2](t) + [theta][I.sub.v2](t) = -[[lambda].sub.0][e.sup.[alpha]t], 0 [less than or equal to] t [less than or equal to] [T.sub.2] (4)

The boundary conditions are

[I.sub.v1](t) = 0, t = 0 (5)

[I.sub.v2](t) = 0, t = [T.sub.2] (6)

[I.sub.b](t) = [I.sub.0], t = 0 (7)

[I.sub.b](t) = 0, t = [t.sub.1] (8)

[I.sub.v1]([T.sub.1]) = [I.sub.v2](0) (9)

And,

T = [T.sub.2]/n (10)

The solutions of the above differential equations obtained are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

Using the condition that we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

If the product of the deterioration rate and the replenishment interval is much smaller than one, the buyer's and the vendor's actual average inventory level, [[bar.I].sub.b] and [[bar.I].sub.v], are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Respectively.

The annual total holding cost for the buyer and the vendor are

[HC.sub.b] = [C.sub.b][C.sub.bh][[bar.I].sub.b] (18)

And

[HC.sub.v] = [C.sub.v][C.sub.vh][[bar.I].sub.v] (19)

Respectively.

The annual deterioration cost for the buyer and the vendor are

[DC.sub.b] = [C.sub.b][theta][[bar.I].sub.b] (20)

And

[DC.sub.v] = [C.sub.v][theta][[bar.I].sub.v] (21)

Respectively.

The annual set-up cost for the buyer and the vendor are

[OC.sub.b] = [C.sub.bs]/T (22)

and

[OC.sub.v] = [C.sub.vs]/[T.sub.2] (23)

Respectively.

The annual shortage cost for the buyer is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

The different costs associated with the system are set-up costs, holding costs, deterioration cost and shortage cost. Our aim is to minimize the total cost.

From (9), one can derive the following condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

By Taylor's series expansion, (25) is derived as

[T.sub.1] = 1/K - 1[T.sub.2][1 + ([alpha] + [theta]/2 [T.sub.2]] (26)

Regarding interest charged and interest earned based on the length of the cycle time [t.sub.1], three cases arise:

Case I: M [greater than or equal to] [t.sub.1]

[FIGURE 1 OMITTED]

In the first case, retailer does not pay any interest to the supplier. Here, retailer sells I0 units during (0, [t.sub.1]) time interval and paying for [CI.sub.0] units in full to the supplier at time M [greater than or equal to] [t.sub.1], so interest charges are zero, i.e.

[IC.sub.1] = 0 (27)

Retailers deposits the revenue in an interest bearing account at the rate of Ie/\$/ year. Therefore, interest earned [IE.sub.1], per year is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

Total cost per unit time of an inventory system is

[TC.sub.b] = [OC.sub.b] + [HC.sub.b] + [DC.sub.b] + [SC.sub.b] + [IC.sub.1] - [IE.sub.1] (29)

[TC.sub.v] = [OC.sub.v] + [HC.sub.v] + [DC.sub.v] - [IC.sub.1] (30)

To minimize the total cost per unit time, the optimum value of [t.sub.1], [T.sub.2] is the solution of following equation.

Case II: M < [t.sub.1]< N

[FIGURE 2 OMITTED]

In the second case, supplier charges interest at the rate [Ic.sub.1] on unpaid balance.

Interest earned, [IE.sub.2] during [0, M] is

[IE.sub.2] = [PI.sub.e][[intergral].sup.M.sub.0][e.sup.-rt]D(t)tdt (31)

Retailer pay for [I.sub.0] units purchased at time t = 0 at the rate of C/\$/ unit to the supplier during [0, M]. The retailer sells D (M).M units at selling price P/unit. So, he has generated revenue of P D(M).M + [IE.sub.2]. Then two sub cases may arise:

Sub Case: 2.1

Let P D(M).M + [IE.sub.2] [greater than or equal to] [CI.sub.0], i.e. retailer has enough money to settle his account for all [I.sub.0] units procured at time t = 0. Then interest charge will be

[IC.sub.2.1] = 0 (32)

And interest earned

[IE.sub.2.1] = [IE.sub.2]/[T.sub.2] (33)

So, total cost [TC.sub.2.1] per unit time of inventory system is

[TC.sub.b] = [OC.sub.b] + [HC.sub.b] + [DC.sub.b] + [SC.sub.b] + [IC.sub.2.1] - [IE.sub.2.1] (34)

[TC.sub.v] = [OC.sub.v] + [HC.sub.v] + [DC.sub.v] - [IC.sub.2.1] (35)

To minimize the total cost per unit time, the optimum value of [t.sub.1], [T.sub.2] is the solution of following equation.

Sub Case: 2.2

Let P D(M).M + [IE.sub.2] < [CI.sub.0]. Here, retailer will have to pay interest on unpaid balance [U.sub.1] = [CI.sub.0] - (P D(M).M + [IE.sub.2]) at the rate of Ic1 at time M to the supplier. Then interest paid per unit time is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (36)

And interest earned

[IE.sub.2.2] = [IE.sub.2]/[T.sub.2] (37)

So, total cost [TC.sub.2.2] per unit time of inventory system is

[TC.sub.b] = [OC.sub.b] + [HC.sub.b] + [DC.sub.b] + [SC.sub.b] + [IC.sub.2.2] - [IE.sub.2.2] (38)

[TC.sub.v] = [OC.sub.v] + [HC.sub.v] + [DC.sub.v] - [IC.sub.2.2] (39)

To minimize the total cost per unit time, the optimum value of [t.sub.1], [T.sub.2] is the solution of following equation.

Case III: [t.sub.1] [greater than or equal to] N

[FIGURE 3 OMITTED]

In the final case, retailer pays interest at the rate of [Ic.sub.2] to the supplier. Based on the total purchased cost, CI0, total money P D(M).M + [IE.sub.2] in account at M and total money P D(N).N + [IE.sub.2] at N, there are three sub cases may arise:

Sub Case 3.1 Let P D(M).M + [IE.sub.2] [greater than or equal to] [CI.sub.0]

This case is same as sub case 2.1, here 3.1 designate decision variables and objective function.

Sub Case 3.2 Let P D(M).M + [IE.sub.2] < [CI.sub.0] and

PD(N - M)*(N - M) + [PI.sub.e][[integral].sup.N.sub.M]D(t)dt[greater than or equal to][CI.sub.0] - (PD(M)*M + [IE.sub.2])

This case similar to sub case 2.2.

Sub Case 3.3 Let P D(M).M + [IE.sub.2] < [CI.sub.0] and

PD(N - M)*(N - M) + [PI.sub.e][[integral].sup.N.sub.M]D(t)dt<[CI.sub.0] - (PD(M)*M + [IE.sub.2])

Here, retailer does not have enough money to pay off total purchase cost at N. He will not pay money of P D(M).M + [IE.sub.2] at M and PD(N - M)*(N - M) + [PI.sub.e][[integral].sup.N.sub.M]D(t)dt at N. That's why he has to pay interest on unpaid balance [U.sub.1] = [CI.sub.0] - (P D(M)*M + [IE.sub.2]) with [Ic.sub.1] interest rate during (M, N) and [U.sub.2] = [U.sub.1] - PD(N - M)*(N - M) + [PI.sub.e][[integral].sup.N.sub.M]D(t)dt with interest rate [Ic.sub.2] during (N, [t.sub.1]).

Therefore, total interest charged on retailer, [Ic.sub.3.3] per unit time is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (40)

Interest earned per unit time is

[IE.sub.3.3] = [IE.sub.2]/[T.sub.2] (41)

So, total cost [TC.sub.3.3] per unit time of inventory system is

[TC.sub.b] = [OC.sub.b] + [HC.sub.b] + [DC.sub.b] + [SC.sub.b] + [IC.sub.3.3] - [IE.sub.3.3] (42)

[TC.sub.v] = [OC.sub.v] + [HC.sub.v] + [DC.sub.v] - [IC.sub.3.3] (43)

To minimize the total cost per unit time, the optimum value of [t.sub.1], [T.sub.2] is the solution of following equation.

Numerical examples

The preceding theory can be illustrated by the following numerical example where the parameters are given as follows:

The preceding theory can be illustrated by the following numerical example where the parameters are given as follows:

Demand parameters, [[lambda].sub.0] = 500, [alpha] = 0.05

Selling price, P = 30

Buyer's purchased cost, [C.sub.b] = 35

Buyer's percentage holding cost per year per dollar, [C.sub.bh] = 0.1

Buyer's ordering cost per order, [C.sub.bs] = 500

Buyer's shortage cost, [S.sub.b] = 50

Vendor's unit cost, [C.sub.v] = 20

Vendor's percentage holding cost per year per dollar, [C.sub.vh] = 0.1

Vendor's setup cost per order, [C.sub.vs] = 500

Vendor's production rate per year, K = 3

Deterioration rate, [theta] = 0.01

First delay period, M= 0.08

Second delay period, N= 0.1

Discount rate, r = 0.12

The interest earned, Ie = 0.05

The interest charged, Ic1 = 0.12

The interest charged, Ic2 = 0.20 (Ic1 > Ic2)

Observation

The data obtained clearly shows that individual optimal solutions are very different from each other. However, there exists a solution which ultimately provides the minimum operating cost to the whole supply chain. All the observations can be summed up as follows:

1. An increase in the interest charged, increases the buyer cost BC and decrease the vendor cost VC of the commodity.

2. Optimal solution for the buyer is n=1 in table first while for the vendor, it is n=5 in table fourth. The overall optimal solution which ultimately minimizes the cost across the whole supply chain is n=5 in table fourth.

Conclusion

In real world, it is noted that, as a result of progressive permissible delay in settling the replenishment account, the economic replenishment interval and order quantity generally increase marginally, although the annul cost decreases considerably. The saving in cost as a result of permissible delay in settling the replenishment account largely come the ability to delay payment without paying any interest. Presence of inflation in cost and its impact on demand suggest larger cycle length. So this EOQ model is applicable when the inventory contains trade credit that supplier give to the retailer.

References

 A.J. Clark, H. Scarf (1960), Optimal policies for a multi-echelon inventory problem, Management Sciences, 6, 475-490

 J.A. Buzacott (1975), Economic order quantities with inflation, Operation research Quarterly, 26, 553-558

 S. Bose, A. Goswami and K.S. Choudhuri (1995), An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting, Journal of the Operational Research society, 27, 213-224

 H. Hwang and S.W. Shinn (1997), Retailer's pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments Computers & Operations Research, Volume 24, 6, 539-547.

 D. Ha, S.L. Kim (1997), Implementation of JIT purchasing an integrated approach, Production Planning & Control Management, 8 (2), 152-157

 B.R. Sarker, A. M. M. Jamal and S. Wang (2000), Supply chain models for perishable products under inflation and permissible delay in payment, Computers & OperationsResearch, 27, 1, 59-75.

 P.C. Yang, H.M. Wee (2000), Ecomonic order policy of deteriorated items for vendor and buyer: An integral approach, Production Planning and control, 11(5), 474-480

 K.J. Chung and Y.F. Huang (2003), The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84, 3, 307-318.

 C.T. Chang (2004), An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity, International Journal of Production Economics, 88, 3, 307-316

 K.J. Chung, S. K. Goyal and Y.F. Huang (2005), The optimal inventory policies under permissible delay in payments depending on the ordering quantity International Journal of Production Economics, 95, 2, 203-213.

 H.L. Yang (2006), Two-warehouse partial backlogging inventory models for deteriorating items under inflation, International Journal of Production Economics, 103, 1, 362-370

 X. Song and X. Cai (2006), On optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Production Economics, 103,1, 246-251.  H.T. Lee and J.C. Wu (2006), A study on inventory replenishment policies in a two - echelon supply chain system, Computers & Industrial Engineering, 51, 2, 257-263

 J.J. Liao (2007), On an EPQ model for deteriorating items under permissible delay in payments, Applied Mathematical Modelling, 31, 3, 393-403.

 L.H. Chen and F.S. Kang (2007), Integrated vendor-buyer cooperative inventory models with variant permissible delay in payments, European Journal of Operational Research, 183, 2, 658-673

 S.T. Lo, H.M. Wee, W.C. Huang (2007), An integrated production-inventory model with imperfect production processes and weibull distribution deterioration under inflation, International Journal of Production Economics, 106, 1, 248-260

Rakesh Agrawal *, Dr. Debangana Rajput and Dr. N.K. Varshney

Deptt. of Mathematics, D.S. College, Aligarh U.P., India
```Table 1

n [T.sub.2] [t.sub.1] VC BC TC

1 0.824546 0.136863 1263.1 102.751 1365.85
2 0.856092 0.138116 1253.91 122.435 1376.34
3 0.886756 0.138522 1246.96 177.757 1424.71
4 0.916491 0.138723 1241.84 229.542 1471.38
5 0.945356 0.138844 1238.25 278.300 1516.55

Table 2

n [T.sub.2] [t.sub.1] VC BC TC

1 0.804371 0.134818 1289.28 117.89 1407.17
2 0.827892 0.135711 1267.23 146.93 1414.16
3 0.845642 0.135993 1254.60 189.22 1443.82
4 0.866030 0.136018 1251.72 235.65 1487.37
5 0.887485 0.136614 1247.44 285.90 1533.34

Table 3

n [T.sub.2] [t.sub.1] VC BC TC

1 0.804371 0.134818 1254.49 140.66 1395.15
2 0.827892 0.135711 1249.70 172.78 1422.48
3 0.845642 0.135993 1237.88 199.60 1437.48
4 0.866030 0.136018 1222.91 245.89 1468.80
5 0.887485 0.136614 1199.45 298.44 1497.89

Table 4

n [T.sub.2] [t.sub.1] VC BC TC

1 1.388450 0.754290 1212.76 179.23 1391.99
2 1.457826 0.568734 1199.89 198.91 1398.80
3 1.583762 0.429980 1172.90 210.44 1385.34
4 1.673398 0.299718 1150.00 255.10 1425.10
5 1.814529 0.157600 1010.27 301.00 1311.27
```
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