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The proposed model considers the products with finite shelf-life which causes a small amount of decay. The market demand is assumed to be level dependent and in a linear form. The model has also considered the constant production rate which stops attaining a desired level of inventories and that is the highest level of inventories. Production starts with a buffer stock and without any sort of backlogs. Due to the market demand and product’s decay, the inventory reduces to the level of buffer stock where again the production cycle starts. With a numerical search procedure the proof of the proposed model has been shown. The objective of the model is to obtain the total average optimum inventory cost and optimum ordering cycle.

With a view to solving the inventory problems, it is highly essential for the business institutions to obtain the economic order quantity (EOQ) and obtaining this quantity leads to reduce the total average inventory cost. This is why the business institutions emphasize on inventory management and solving inventory problems. The problem can only be solved if a suitable inventory model could be established which is fit for all the parameters concerned like, market demand, production rate, product’s life, etc. The innovative EOQ model is, therefore, a highly demand on regular basis and when required in spite of having existence of huge number of inventory models. Inventory, indeed, is a stock of materials. Inventory problems are mainly related to the proper management of this inventory which can lead to minimize the inventory cost. Generally, we have two kinds of materials in our daily needs as far as damage, wastage, deterioration or decay is concerned. Items like radioactive substances, food grains, fashionable items, pharmaceuticals, etc. are the items of finite life and the items like electronic goods, steels, woods, etc. are the items of ling life. Due to the limited shelf-life and market demand, the stock level or inventory continuously decreases and the items in the inventory deplete or deteriorate. This deterioration affects the inventory 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 to be formulated. An inventory model with linear demand, small amount of decay and constant production rate has been proposed in this paper to minimize inventory cost. Keeping the buffer stock as a reserve, the production is assumed to start and after certain periods at the highest level of inventory, it stops. In this model, we have considered constant production rate along with the deterioration, whereas the classical inventory models and many researchers use the instantaneous replenishment. Finally, by proving the convex property and using a numerical searcher procedure, the paper justified the correctness of the model.

Since long the researchers had been focusing on obtaining inventory models suitable to the needs in real life with a view to solving inventory problems. The problems are related with what will be the pattern of demand in the market, what may be the production rate of the business institutions, whether there will be finite life of the products, whether backlogs or shortages and delay in payments are allowed etc. Many researchers have structured various types of inventory model basing on the situation or the market demand. It may arise different types of demands in the market. Demand may be linear, quadratic, exponential, time dependent, level or stock dependent, price dependent etc. Considering all the parameters the inventory model is designed. There are two types of models in this field which covers all the parameters mentioned above. One is deterministic model which deals with the constant demand and lead time; the other one is stochastic or probabilistic model which deals randomly with the variable demand and lead time. In this review of the literature, mostly the inventory models with deterministic demand have been discussed. Determining EOQ is one of the most important factors to formulate the inventory model. The ultimate aim of formulating the model is to minimize the inventory cost by finding the EOQ. Harris [

・ Production rate

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・ For unit inventory, amount of decay rate

・ Production starts with a few amounts of items in the inventory as a buffer stock.

・ Inventory level is highest at a specific level and after this point, the inventory depletes quickly due to demand and deterioration.

・ Shortages are not allowed.

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Basing on the demand pattern, the business institution decides the structure of the model (

The inventory increases at the rate of

below:

The general solution of the differential equation is

Applying the following boundary condition, we get

Therefore,

From the other boundary condition, i.e. at

Using the Equation (1) and considering up to second degree of

We calculate the deteriorating items during the period considering the decay of the items as below:

On the other hand, the inventory decreases at the rate of

The general solution of the differential equation is defined below:

Applying the boundary condition at

By solving we get,

Therefore,

Substituting another boundary condition, i.e. at

Now, using Equation (5) and considering up to the first 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,

Or,

Considering the value as

We construct the following equation with the help of Equation (9),

Total Cost Function: The cost function can be described in the following form,

By substituting the Equations (3), (4), (7), (8) and (11) in (12), we get the value of total average inventory cost as below,

Now with a view to obtaining the total time cycle

(i)

Now differentiating Equation (13) with respect to

Putting the value of Equation (14) in the convex property (i) and then using (10), we get

Or,

Or,

Now with the help of Equations (11) and (15), we get the value of

Again differentiating Equation (14) with respect to

From Equation (17) we come to an end that the convex property (ii) is satisfied, i.e.

According to the result in section 5, we give an example that may illustrate how the numerical search procedure works. Suppose that there is a product which is a linear function in the inventory system and adopts the following parameters:

We now put all the values in Equations (15), (16), (2) and (13) and then we get the results as optimum order interval

Now, how the inventory system or the solution is affected by even a little changes of parameters

Order Interval ( | 5.250 | 5.500 | 5.750 | 6.000 | 6.252 | 6.500 | 6.750 | 7.000 | 7.250 |
---|---|---|---|---|---|---|---|---|---|

Total Cost ( | 52.67 | 52.45 | 52.30 | 52.21 | 52.19 | 52.21 | 52.28 | 52.39 | 52.54 |

Parameters | Change in % | Value of | |||
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+50 | 1.736 | 5.061 | 68.808 | 70.696 | |

+25 | 1.685 | 5.570 | 69.625 | 61.395 | |

+10 | 1.656 | 5.954 | 70.115 | 55.852 | |

−10 | 1.620 | 6.592 | 70,769 | 48.562 | |

−25 | 1.595 | 7.200 | 71.259 | 43.205 | |

−50 | 1.555 | 8.591 | 72.076 | 34.569 | |

+50 | 1.265 | 7.241 | 111.392 | 48.121 | |

+25 | 1.416 | 6.754 | 90.918 | 49.900 | |

+10 | 1.537 | 6.454 | 78.632 | 51.205 | |

−10 | 1.762 | 6.052 | 62.252 | 53.265 | |

−25 | 2.014 | 5.765 | 49.967 | 54.996 | |

−50 | 2.884 | 5.505 | 29.492 | 56.826 | |

+50 | 1.697 | 5.438 | 66.347 | 57.338 | |

+25 | 1.667 | 5.807 | 68.395 | 54.747 | |

+10 | 1.649 | 6.063 | 69.623 | 53.200 | |

−10 | 1.627 | 6.457 | 71.261 | 51.192 | |

−25 | 1.611 | 6.797 | 72.490 | 49.727 | |

−50 | 1.582 | 7.477 | 74.537 | 47.378 | |

+50 | 1.735 | 5.072 | 63.890 | 60.491 | |

+25 | 1.684 | 5.577 | 67.166 | 56.293 | |

+10 | 1.656 | 5.957 | 69.132 | 53.812 | |

−10 | 1.621 | 6.588 | 71.753 | 50.602 | |

−25 | 1.595 | 7.186 | 73.7187 | 48.302 | |

−50 | 1.556 | 8.549 | 76.994 | 44.752 | |

+50 | 1.337 | 5.105 | 70.442 | 70.284 | |

+25 | 1.465 | 5.592 | 70.442 | 61.237 | |

+10 | 1.562 | 5.961 | 70.442 | 55.808 | |

−10 | 1.727 | 6.590 | 70.442 | 48.570 | |

−25 | 1.891 | 7.219 | 70.442 | 43.142 | |

−50 | 2.317 | 8.842 | 70.442 | 34.095 | |

+50 | 1.635 | 6.217 | 70.360 | 52.469 | |

+25 | 1.636 | 6.238 | 70.409 | 52.301 | |

+10 | 1.637 | 6.245 | 70.426 | 52.245 | |

−10 | 1.638 | 6.259 | 70.459 | 52.133 | |

−25 | 1.639 | 6.266 | 70.475 | 52.078 | |

−50 | 1.640 | 6.287 | 70.524 | 51.910 |

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Because of the development of inventory management in the present age, the business institution cannot think its cost minimization without the proper use of it. By the proper use, management and thereby developing the suitable inventory models, the business enterprise can save its huge inventory cost. Before using model the enterprise needs to know the actual pattern of demand in the market. This demand always fluctuates. The suitable model is developed by considering the actual demand. The inventory model we have proposed in this paper is dependent on the stock, even we have considered buffer stock. Hence, the stock goes out due to any unavoidable circumstances, demand could still be met. The model also considers the deterioration, so due to the finite shelf-life of the items this model gives the correct. In the proposed model, the production rate and the decay have been considered constant through. The model develops an algorithm to determine the optimum ordering cost, total average optimum inventory cost, optimum time at maximum inventory level and optimum time cycle. The model could establish that with a particular order level

The authors thank the editor and the reviewers for their valuable comments which could play a significant role to improve the standard of the manuscript.

ShirajulIslam Ukil,Md. SharifUddin, (2016) A Production Inventory Model of Constant Production Rate and Demand of Level Dependent Linear Trend. American Journal of Operations Research,06,61-70. doi: 10.4236/ajor.2016.61008