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In the classical inventory models, it is assumed that the retailer pays to the supplier as soon as he received the items and in such cases the supplier offers a cash discount or credit period (permis-sible delay) to the retailer. In this paper we presented an inventory model for perishable items with time varying stock dependent demand under inflation. It is assumed that the supplier offers a credit period to the retailer and the length of credit period is dependent on the order quantity. The purpose of our study is to minimize the present value of retailer’s total cost. Numerical examples are also given to demonstrate the presented mode.

In the classical inventory models payment for the items paid by the supplier depends on the payment paid by the retailer and in such cases the supplier offers a fixed credit period to the retailer during which no interest will be charged by the supplier so there is no need to pay the purchasing cost by the retailer and after this credit period up to the end of a period interest charged and paid by the retailer. In such situations the retailer starts to accumulate revenue on his sale and earn interest on his revenue. If the revenue earned by the retailer up to the end of credit period is enough to pay the purchasing cost or there is a budget then the balance is settled and the supplier does not charge any interest, otherwise the supplier charges interest for unpaid balance after the credit period. The interest and the remaining payment are made at the end of replenishment cycle.

In traditional EOQ models the payment time does not affect the profit and replenishment policy. If we consider the inflation then order quantity and payment time can influence both the supplier’s and retailer’s decisions. A large pile of perishable foods such as fruits, vegetables, milk, bread, chocklet etc. attract the consumers to buy more. Buzacott [

In the present paper we presented an inventory model for perishable items with time varying stock dependent demand and trade credit under inflation. Although there are so many research papers related to the perishable

1.2 | 0.809153 | |

1.4 | 1.29391 | |

1.6 | 1.70349 | |

1.8 | 2.07928 | |

2 | 2.43731 | |

2.2 | 2.78705 | |

2.5 | 3.30941 |

2 | 0.809153 | |

4 | 3.71242 | |

6 | 6.66987 | |

8 | 9.64061 | |

10 | 12.6158 | |

12 | 15.5929 |

0.05 | 0.494393 | |

0.1 | 0.500148 | |

0.15 | 0.500591 | |

0.2 | 0.495351 | |

0.25 | 0.483842 | |

0.3 | 0.465167 | |

0.35 | 0.437937 | |

0.4 | 0.399872 |

2 | 0.494393 | |

4 | 1.02028 | |

6 | 1.54603 | |

8 | 2.07174 | |

10 | 2.59744 | |

12 | 3.12314 |

products with stock dependent demand under inflation. This paper deals with the same type problem and it provides an approximate solution procedure of this problem for minimizing the present value of retailer’s total cost.

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

1) The demand rate

2)

3)

4)

5)

6) M is the credit period.

7) T is the replenish cycle length.

8) r is the inflation rate.

9)

10) C is the purchasing cost per unit.

11) P is the selling price per unit with

12) Q is the initial inventory level.

13) L is the planning horizon.

14) The supplier sells one single item to the retailer.

15) The items are replenished when the stock level becomes zero.

16) The supplier provides a credit period, which is dependent on the order quantity.

17) The lead time is zero.

18) Shortages are not allowed.

19) The inventory planning horizon is finite and the numbers of cycles are finite in the planning horizon.

20) I(t) is the inventory level at any time t.

Suppose an inventory system consists the maximum inventory level at any time t = 0 and due to both demand and deterioration the inventory level decreases in the interval

With the boundary condition

The equation (1) can also be written as

where

With the boundary condition

For a 2^{nd} order approximation of

Using the boundary condition, I(0) = Q the initial order quantity is

Now we discuss the following two cases

(1) and (2)

When

During the 1st cycle the present value of ordering cost is

During the 1st cycle the present value of purchasing cost is

During the 1st cycle the present value of holding cost is

Therefore during the 1st cycle the present value of retailer’s total cost is

Since there are m cycles in the planning horizon L then the present value of retailer’s total cost over the planning horizon L is

The necessary condition for

When

1) Let

so in this case no interest will be charged by the supplier although the credit period M is smaller than the replenishment cycle length T so the present value of retailer’s total will be same as that in case I.

2) Let

and the retailer has a budget to pay the remaining short purchasing cost so in this case there is still no interest charged by the supplier although the credit period M is smaller than the replenishment cycle length T so the present value of retailer’s total will be same as that in case I.

3) Let

and the retailer has no budget to pay the remaining short purchasing cost so in this case for unpaid balance the interest will be charged by the supplier from M to T. The interest and the remaining payments are made at the end of replenishment cycle. So in this case the retailer’s total cost containing the ordering cost, holding cost, purchasing cost paid at M, the interest and the remaining payments are made at the end of replenishment cycle.

The present values of retailer’s ordering and holding are same cost as in case I

During the first cycle the purchasing cost paid at M is equal to the amount of revenue earned by the retailer up to M so

During the first cycle the present values of remaining payments and interest paid at the end of replenishment cycle are

During the 1st cycle the present value of retailer’s total cost is

Since there are m cycles in the planning horizon L then the present value of retailer’s total cost over the planning horizon L is

The necessary condition for

Let us consider the following parameters in the appropriate units

When

Since

As we increase the parameter r then the value of total cost decreases.

As we increase the parameter M then the value of total cost decreases.

Let us consider the following parameters in the appropriate units

When

As we increase the parameter r then the value of total cost increases.

As we increase the parameter M then the value of total cost decreases.

In this paper, we proposed an inventory model for perishable items with time varying stock dependent demand under inflation and time discounting. In the numerical analysis we study the effect of the change of the parameters r and M on the optimal solution. From

SushilKumar,U. S.Rajput, (2015) An Inventory Model for Perishable Items with Time Varying Stock Dependent Demand and Trade Credit under Inflation. American Journal of Operations Research,05,435-449. doi: 10.4236/ajor.2015.55036