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A production inventory model has been developed in this paper, basing on constant production rate and market demand, which varies time to time. Seeing the demand pattern the proposed model has been formulated in a power pattern which can be expressed in a linear or exponential form. The model finds the total average optimum inventory cost and optimum time cycle. The model also considers the small amount of decay. Without having backlogs, production starts. Reaching at the desired level of inventories, it stops production. After that due to demands along with the deterioration, it initiates its depletion and after certain periods the inventory gets zero. The model has also been justified with proving the convex property and by giving a numerical example with the sensitivity test.

In last few decades, inventory problems have been studied in a large scale. Inventory is related to stock the items before delivering to the customers. If we visualize our daily needs, we can see that these needs are of two kinds as far as deterioration or decay is concerned. Items like radioactive substances, food grains, fashionable items, pharmaceuticals etc. are one kind, which have finite shelf-life, i.e. limited life once in the shelf, or have a sufficient deterioration after a particular time. On the other hand, the items like bricks, steels, heavy woods etc. are the other kind which does not have that much of deterioration the one mentioned before. Due to the limited shelf-life and market demand, the stock level or inventory continuously decreases and in this way deterioration occurs. This deterioration affects the inventory seriously and inventory cost increases. To make the inventory cost at optimum level i.e. to get the minimum inventory cost, a suitable inventory model is required which suits to meet the actual demand in the market. In minimizing inventory cost this paper proposes an inventory model with power demand, small amount of decay and constant production rate, whereas the existing models very often ignore the production rate; instead those consider the instantaneous replenishment rate. The power demand defines that kind of demand which varies with change of power in the power function.

The organizations give due importance to few parameters which affect the model. Like, in this proposed model power demand has been considered as the market may have a demand of linear type, again shortly it may have the demand of exponential function. The linear demand means that the firms receive demand either in an increasing or decreasing way, but gradually, not suddenly, i.e. demand as a linear function. Again, the exponential demand means the demand either in an exponentially increasing way or exponentially decreasing way suddenly. It may be clarified more giving an example, like,

Satisfying the convex property and using a numerical example, the paper could justify that the objective of formulating this model is achieved. The objective of the proposed model is to get the optimum inventory cost and optimum time cycle by introducing a time dependent inventory model with constant production rate and power demand. The paper subsequently advances with literature review, assumptions, notations used in the model, development of the model, numerical illustration, sensitivity analysis, conclusion and suggestions for future work in this field.

Many researchers have work in the field of inventory problems or in production inventory model to solve the real life problems by building the suitable inventory models. On ground, the business institution faces various types of demand. Demand may be linear, quadratic, exponential, time dependent, level or stock dependent, price dependent etc. Basing on the demand pattern, the firms decide how much to produce and when to produce. Initially, Harris [

To develop the proposed model it needs numbers of notations or symbols to clarify the assumptions considered and description explained in this paper. The notations or symbols used in this paper are cited in

There may be various types of demand of different types of items in the market. At times it may be linear. Again within vary short span of time it may be changed into exponential type of demand. Taking this type of situation into cognizance, this model is developed. The model is suitable for those kinds of products which have finite shelf-life and ultimately causes the products decay. At the beginning, while time

Ser | Notations | Description |
---|---|---|

1 | Production rate. | |

2 | Demand rate at time T where | |

3 | Very small amount of constant decay rate for unit inventory. | |

4 | Inventory level at instant | |

5 | Un-decayed inventory at | |

6 | Un-decayed inventory at | |

7 | Deteriorating inventory at | |

8 | Deteriorating inventory at | |

9 | Inventory level at time T which depicts 0 and | |

10 | Vary small portion of instant | |

11 | Set up cost. | |

12 | Average holding cost. | |

13 | Total cost in terms of | |

14 | Optimum order quantity. | |

15 | Optimum order interval. |

During the period,

The general solution of the differential equation is,

We now apply the following boundary condition, at

By solving we get,

Therefore,

Putting another boundary condition, i.e. at

Thereby,

With the help of Equation (1) and by considering up to second degree of

Considering the decay of the items, we calculate the deteriorating items during the period as below:

Again during

The general solution of the differential equation is defined below:

Applying the boundary condition at

By solving we get,

Therefore,

Putting another boundary condition, i.e. at

Thereby,

Now, with the help of Equation (5) and by considering up to the second degree of

Considering the decay of the items, we calculate the deteriorating items during the period as below:

From Equations (2) and (6) we get,

Therefore,

Total Cost Function: The cost function can be written in the form given below:

By using Equations (3), (4), (7) and (8) in (10), we get the following result,

Now using Equations (2) and (9) we get the value as,

The objective is now to find out the order quantity

1)

2)

From the convex property 1) i.e.

or,

Therefore, the optimum order quantity,

Now again differentiating the Equation (11) with respect to

From Equation (13) we can conclude that the convex property 2) i.e.

Finally, we can conclude that the equation of total cost function (11) is convex in

1) Situation 1: When n = 1 (Demand is Linear Function)

In order to give an example with numerical illustration, let us suppose the following parameters, while

After putting all the values in Equations (2), (9), (11) and (12) we get the following results:

・ Optimum time

・ Optimum order interval

・ Total average optimum inventory cost

・ Optimum order quantity

Putting the values of

It can be mentioned that

2) Situation 2: When n = 2 (Demand is Exponential Function)

In order to give an example with numerical illustration, let us suppose same parameter we have considered in the first situation as below while

After putting all the values in equation no (2), (9), (11) and (12) we get the following results:

・ Optimum time

・ Optimum order interval

・ Total average optimum inventory cost

・ Optimum order quantity

Putting the values of

In this case, it is observed that

Now assuming the set up cost

Inventory (Q_{1}) | 35.210 | 37.210 | 39.210 | 41.210 | 43.210 | 45.210 | 47.210 |
---|---|---|---|---|---|---|---|

Total cost (TC) | 8.844 | 8.781 | 8.747 | 8.736 | 8.745 | 8.773 | 8.817 |

Remarks | At a particular inventory level total cost is minimum, before and after this point total cost increases |

Inventory (Q_{1}) | 30.71 | 33.71 | 36.71 | 39.71 | 42.71 | 45.71 | 48.71 |
---|---|---|---|---|---|---|---|

Total cost (TC) | 16.65 | 16.34 | 16.17 | 16.12 | 16.16 | 16.28 | 16.46 |

Remarks | At a particular inventory level total cost is minimum, before and after this point total cost increases |

Parameters | Change in % | Value of | |||
---|---|---|---|---|---|

+50 | 1.472 | 22.077 | 50.512 | 27.545 | |

+25 | 1.792 | 22.397 | 46.190 | 8.019 | |

+10 | 2.061 | 22.666 | 43.295 | 8.409 | |

−10 | 2.576 | 23.180 | 38.980 | 9.136 | |

−25 | 3.170 | 23.775 | 35.304 | 9.937 | |

−50 | 5.150 | 25.756 | 27.905 | 12.350 | |

+50 | 2.424 | 16.161 | 40.694 | 12.534 | |

+25 | 2.355 | 18.839 | 41.054 | 10.657 | |

+10 | 2.315 | 21.047 | 41.183 | 9.509 | |

−10 | 2.264 | 25.159 | 41.166 | 7.958 | |

−25 | 2.228 | 29.701 | 40.888 | 6.787 | |

−50 | 2.169 | 43.379 | 39.146 | 4.860 | |

+50 | 2.424 | 16.161 | 40.694 | 12.534 | |

+25 | 2.355 | 18.839 | 41.054 | 10.657 | |

+10 | 2.315 | 21.047 | 41.183 | 9.509 | |

−10 | 2.264 | 25.159 | 41.166 | 7.958 | |

−25 | 2.228 | 29.701 | 40.888 | 6.787 | |

−50 | 2.169 | 43.379 | 39.146 | 4.860 | |

+50 | 2.289 | 22.894 | 33.648 | 10.920 | |

+25 | 2.289 | 22.894 | 36.859 | 9.828 | |

+10 | 2.289 | 22.894 | 39.292 | 9.172 | |

−10 | 2.289 | 22.894 | 43.434 | 8.299 | |

−25 | 2.289 | 22.894 | 47.585 | 7.644 | |

−50 | 2.289 | 22.894 | 58.280 | 6.552 | |

+50 | 2.289 | 22.894 | 40.502 | 8.890 | |

+25 | 2.289 | 22.894 | 40.934 | 8.795 | |

+10 | 2.289 | 22.894 | 41.073 | 8.765 | |

−10 | 2.289 | 22.894 | 41.345 | 8.707 | |

−25 | 2.289 | 22.894 | 41.476 | 8.680 | |

−50 | 2.289 | 22.894 | 41.855 | 8.602 | |

+50 | 2.400 | 16.970 | 40.833 | 11.895 | |

+25 | 2.339 | 19.666 | 41.117 | 10.193 | |

+10 | 2.308 | 21.533 | 41.197 | 9.291 | |

−10 | 2.273 | 24.356 | 41.190 | 8.215 | |

−25 | 2.249 | 26.753 | 41.094 | 7.497 | |

−50 | 2.217 | 32.357 | 40.738 | 6.452 |

Analyzing the results in the above table we can summarize the following observations:

1)

2)

3)

4)

Analyzing the results in

Parameters | Change in % | Value of | |||
---|---|---|---|---|---|

+50 | 1.527 | 11.453 | 50.382 | 14.154 | |

+25 | 1.891 | 11.817 | 45.387 | 14.939 | |

+10 | 2.206 | 12.132 | 42.077 | 15.582 | |

−10 | 2.836 | 12.762 | 37.173 | 16.775 | |

−25 | 3.610 | 13.356 | 32.991 | 18.089 | |

−50 | 6.618 | 16.544 | 24.412 | 22.034 | |

+50 | 2.836 | 9.454 | 37.294 | 22.568 | |

+25 | 2.647 | 10.588 | 38.547 | 19.465 | |

+10 | 2.545 | 11.569 | 39.256 | 17.486 | |

−10 | 2.421 | 13.450 | 40.129 | 14.713 | |

−25 | 2.336 | 15.571 | 40.694 | 12.536 | |

−50 | 2.206 | 22,058 | 41.210 | 8.742 | |

+50 | 3.610 | 8.021 | 33.119 | 30.385 | |

+25 | 2.888 | 9.240 | 36.969 | 23.305 | |

+10 | 2.620 | 10.823 | 38.740 | 18.946 | |

−10 | 2.369 | 14.624 | 40.481 | 13.417 | |

−25 | 2.237 | 19.884 | 41.168 | 9.707 | |

−50 | 2.090 | 41.795 | 39.146 | 4.854 | |

+50 | 2.482 | 12.408 | 32.419 | 20.148 | |

+25 | 2.482 | 12.408 | 35.514 | 18.134 | |

+10 | 2.482 | 12.408 | 37.858 | 16.925 | |

−10 | 2.482 | 12.408 | 41.853 | 15.313 | |

−25 | 2.482 | 12.408 | 45.848 | 14.104 | |

−50 | 2.482 | 12.408 | 56.152 | 12.089 | |

+50 | 2.482 | 12.408 | 39.526 | 16.192 | |

+25 | 2.482 | 12.408 | 39.636 | 16.147 | |

+10 | 2.482 | 12.408 | 39.671 | 16.133 | |

−10 | 2.482 | 12.408 | 39.739 | 16.105 | |

−25 | 2.482 | 12.408 | 39.771 | 16.092 | |

−50 | 2.482 | 12.408 | 39.864 | 16.055 | |

+50 | 3.309 | 8.272 | 34.578 | 28.029 | |

+25 | 2.769 | 9.787 | 37.733 | 21.531 | |

+10 | 2.577 | 11.219 | 39.030 | 18.138 | |

−10 | 2.404 | 13.806 | 40.248 | 14.292 | |

−25 | 2.312 | 16.350 | 40.833 | 11.899 | |

−50 | 2.206 | 22.058 | 41.210 | 8.742 |

1)

2)

3)

4)

In the present context of modern age, without the inventory management, the business institution cannot think ahead. By the proper management and thereby developing the suitable inventory model, the institution only can save its production inventory cost. Market demand is always fluctuate. The model is developed considering this demand. Maybe, today the market demand is very high and tomorrow it is low. The inventory model we have proposed in this paper can be suitable to meet both the demands linear or exponential. Because of deterioration, this model also gives the correct result in which the materials have the finite shelf-life. In the proposed model, the production rate and the decay have been considered constant all through. The model develops an algorithm to determine the optimum ordering cost, total average inventory cost and optimum time cycle. The model could establish that with a particular order level the inventory cost is minimal. Here, for n = 1, we got

Shirajul IslamUkil,MohammadEkramol Islam,Md. SharifUddin, (2015) A Production Inventory Model of Power Demand and Constant Production Rate Where the Products Have Finite Shelf-Life. Journal of Service Science and Management,08,874-885. doi: 10.4236/jssm.2015.86088