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This paper describes a deteriorating inventory model with ramp-type demand pattern under stock-dependent consumption rate. The deterioration of the product is considered as probabilistic to make the research a more realistic one. The proposed model assumes partially backorder rate which follows a negative exponential with the waiting time. The effect of inflation and time value of money are incorporated into the model. The purpose of this study is to develop an optimal replenishment policy so that the total profit is maximized. We provide a simple solution procedure to obtain the optimal solutions. Numerical examples along with graphical representations are provided to illustrate the model. Sensitivity analysis of the optimal solution with respect to key parameters of the model has been carried out and the implications are discussed.

In reality, deterioration of items during storage period is a realistic phenomenon in many inventory sectors. Controlling and regulating the deteriorating items are very difficult in practice. In storage system, fruits, vegetables, foodstuffs, etc. deteriorate during their normal storage period. The deteriorating items cannot be used for its original purpose. The loss of inventory due to deterioration cannot be ignored. Thus, it is very essential to control the deterioration of items. A model with exponentially decaying inventory was initially proposed by Covert and Philip [

In classical inventory models, it is often assumed that shortages are either completely backlogged or completely lost. But in the real life, when shortages occur, it is observed that some customers may prefer their demands to be backordered, and some may refuse the backorder case. In this direction, Deb and Chaudhuri [

Classical inventory model considers constant demand rate. However it is observed that the demand rate for electronic goods (e.g., hard disk, RAM, processor, mobile, etc.), new brand of consumer goods, seasonal products (fruits, e.g., mango, orange, etc.) increases linearly at the beginning up to a certain moment as time increases and then stabilizes to a constant rate until the end of the inventory cycle. To represent such type of demand pattern, the “term/ramp-type” is used. Mandal and Pal [

The effects of inflation and time-value of money cannot be ignored for the present study. Several researchers have examined the inflationary effect on the inventory policy. Buzacott [

This model is developed for deteriorating items with ramp-type demand under stock-dependent demand. In addition, different types of probabilistic deteriorations are considered in this model. Shortages are allowed which are backlogged. The effect of inflation and time value of money are incorporated into the model. The main purpose of this paper is to develop an optimal replenishment policy which maximizes the total profit per unit time. The necessary and sufficient conditions of the existence and the uniqueness of the optimal solutions are also provided. Sensitivity analysis of the optimal solution with respect to major parameters and their discursion is carried.

To derive the model, following notation and assumptions are made:

Q order quantity per cycle (units)

r discount rate representing the time-value of money

i inflation rate per unit time

s selling price per unit ($/unit)

C_{a} ordering cost per order ($/order)

C_{h} unit inventory holding cost per week ($/unit/week)

C_{p} purchasing cost per unit purchase ($/unit)

C_{b} backorder cost per unit backorder ($/unit)

C_{l} lost sell cost per unit ($/unit)

I(t) on-hand inventory level at time t

t_{1} length of time in which the inventory level falls to zero (week)

T fixed length of each ordering cycle (week)

1) The model is considered for a single item.

2) Deterioration rate

3) The demand rate D(t) is assumed to be a ramp-type function of time, i.e.,

where

4) S(t) is the selling rate at time t, and it is influenced by the demand rate and the on-hand inventory according to relation

where

5) Shortages are allowed and partially backlogged at a rate

6) The effects of inflation and time-value of money are considered.

7) Lead time is assumed as negligible.

The model considers an inventory model for deteriorating items with ramp-type demand and stock-dependent selling rate. The replenishment at the beginning of the cycle brings the inventory level up to

The solutions of these differential equations depend on the selling rate. There are two cases considering in this paper: (a)

In this case, the selling rate S(t) is

(1) and (2) are in the form

Solving (3) to (5), we obtain

Using the boundary condition

Considering the continuity of

Now the order quantity

The total cost per cycle consists of the following four values

(a) Ordering cost per cycle

(b) Purchase cost per cycle

(c) Holding cost per cycle

(d) Backlogging cost per cycle

(e) Lost sale cost per cycle

(f) Sale revenue per cycle

Therefore, the total profit per unit time under the effect of inflation and time-value of money is

Our objective is to obtain the optimal value of

In this case, the selling rate S(t) is

Hence, (1) and (2) reduce to the following equations

Solving Equations (8) to (10) with the boundary conditions, we obtain

Considering the continuity of

Putting

Now the order quantity

The total cost per cycle consists of the following four values

(a) Ordering cost per cycle

(b) Purchase cost per cycle

(c) Holding cost per cycle

(d) Backlogging cost per cycle

(e) Lost sale cost per cycle

(f) Sale revenue per cycle

Total profit per unit time under the effect of inflation and time-value of money is

Our objective is to find the optimal value of

The total profit function of the system over

It is easy to check that this function is continuous at

In this section, we derive results which ensure the necessary and sufficient conditions of the existence and uniqueness of the optimal solution to maximize the total profit.

From (7), for

where

On the other hand we have

and

Taking first order derivative of

Now if

Has a unique root

From (13), for

where

The above analysis shows that two functions

Now if

To derive the optimal solution, we solve two examples that consist of the different situation of the ramp-type demand and the deterioration rates. Let us consider the following parametric values:

We assume that

Now examine whether the optimal solution is unique.

Hence

We assume that

Now examine whether the optimal solution is unique.

Hence

From above numerical examples we can conclude that the optimal total profit is maximum when μ = 0.7 i.e., for Model I. Now we consider different continuous probabilistic deterioration functions. Based on that, we have done our numerical experiments with the same parametric values as in Example 1.

Here we consider

Here we consider

Here we consider

Here we consider

The graphical representation of Examples 3, 4, 5, and 6 are depicted in

We now study the effects of changes in parameters such as

From

From the above table we can conclude that

In this section, we will discuss some special cases that influence the total profit.

Parameters | Changes in percentage | Changes in total cost for Model I | Changes in total cost for Model II |
---|---|---|---|

−50% | +03.97 | +05.79 | |

−25% | +01.98 | +02.50 | |

+25% | −01.98 | −02.89 | |

+50% | −03.97 | −05.79 | |

−50% | +09.87 | +11.34 | |

−5% | +04.19 | +05.02 | |

+25% | −03.18 | −04.05 | |

+50% | −05.65 | −07.35 | |

−50% | +11.01 | +10.01 | |

−25% | +04.76 | +04.18 | |

+25% | −03.71 | −03.15 | |

+50% | −06.65 | −05.61 | |

−50% | +00.27 | +00.23 | |

−25% | +00.14 | +00.12 | |

+25% | −00.13 | −00.12 | |

+50% | −00.27 | −00.23 | |

−50% | +233.01 | +229.91 | |

−25% | +111.08 | +114.57 | |

+25% | −110.43 | −113.95 | |

+50% | −220.35 | −227.36 | |

−50% | −290.31 | −299.68 | |

−25% | −145.38 | −150.07 | |

+25% | +145.94 | +150.62 | |

+50% | +229.59 | +301.87 | |

−50% | +02.02 | +02.45 | |

−25% | +00.97 | +01.19 | |

+25% | −00.91 | −01.13 | |

+50% | −01.76 | −02.21 | |

−50% | −01.02 | −01.28 | |

−25% | −00.52 | −00.65 | |

+25% | +00.55 | +00.68 | |

+50% | +01.12 | +01.39 |

The necessary condition for

The necessary condition for

The necessary condition for

We use the same parametric values as in Example 2 and we obtain the results for special cases which is listed out in

Special cases | Time (week) | Cost ($/week) |
---|---|---|

Case 1 | 0.5918 | 622.692 |

Case 2 | 0.6054 | 649.811 |

Case 3 | 0.5953 | 654.853 |

In this marketing environment, when a new brand of consumer goods is launched, the demand of goods increases quickly to a certain moment and after some time it stabilizes. Finally, it becomes almost constant. Keeping this type of demand pattern in mind, we considered demand as a ramp-type function of time. To make the research a more realistic one, four different types of continuous probabilistic deterioration functions are considered here. The associated profit function was maximized at the optimal values of decision variables. A unique solution procedure was provided as an optimal solution. Some numerical examples, graphical representations, special cases, and sensitivity analysis were given to illustrate the model. There are several extensions of this work that can constitute future research related in this field. This model can be extended in several ways, like multi-item inventory models, and reliability of the items. Another interesting idea is to consider fuzzy demand case.

The authors would like to thank the reviewers for their helpful comments to improve the paper. The authors are grateful to Guest Editor Professor S. S. Sana for his useful comments. This work was supported by the research fund of Hanyang University (HY-2014-N, Project number 201400000002202) for new faculty members.