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In this paper a time dependent inventory model is developed on the basis of constant production rate and market demands which are exponentially decreasing. It advances in quest of total average optimum cost considering those products which have finite shelf-life. The model also considers the small amount of decay. Without having any sort of backlogs, production starts. Reaching at the desired level of inventories, it stops production. After that due to demands along with the deterioration of the items it initiates its depletion and after certain periods the inventory gets zero. The decay of the products is level dependent. The objective of this paper is to find out the optimum inventory cost and optimum time cycle. The model has also been justified with proving the convex property and by giving a numerical example.

Inventory is an important ingredient of any business. It is a process and place by “proper and in time” utilization of which an enterprise can save a certain amount of production cost and the inventory cost has a vital role to reduce the production cost. Unless inventories are controlled, they are unreliable, inefficient and less cost effective. During our daily lives, generally we come across two types of materials; one is perishable items or items of

having decay and the other one is non perishable items or items of having decay or deterioration. Most of the necessary goods like cosmetics items, radio-active substances, fashion goods, pharmaceuticals, food items etc. decrease due to its finite or limited shelf-life. Due to the limited shelf-life and market demand, the stock level or inventory continuously decreases and thereby, deterioration occurs. To get the actual inventory cost, this deterioration must be taken into consideration. On the other hand, 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 a time dependent inventory model with constant production rate and exponential demand of materials with small amount of decay, whereas the existing models very often ignore the production rate, instead those consider the instantaneous replenishment rate. As the production reaches at a certain level of inventory, the production stops. 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 exponential declining demand. The paper subsequently is unfolded 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.

Sufficient numbers of works have already been done by a large number of researchers in the area of production inventory model to build the suitable inventory models. In last few years, many researchers have studied in this field and developed inventory models to solve the real life problems. In classical inventory models, the demand rate But on ground, it must not be always correct. There may be various type of demands. 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. Harris [

a. Production rate is constant at any time.

b. Production starts when inventory level is zero and it stops when inventory level is highest.

c. Inventory level is highest at

d. Demand rate exponentially decreases.

e. Deteriorating or decay rate is constant and very small.

f. Decreasing rate of demand is also constant and less than decay rate for unit inventory

g. Shortages are not allowed.

h. Lead time is zero.

The model is developed on the basis of exponential market demands and constant production capacity of the organization. The model is suitable for the products which have finite shelf-life and ultimately causes the products decay. At the beginning, i.e. at time

During time

The general solution of the differential equation is,

We now apply the following boundary condition, at

By solving we get,

Therefore,

Applying the other boundary condition, we get

Hence, the total un-decayed inventory during

Neglecting the higher power of

Again during

Applying the boundary condition at

We get the following conditions, if we put the other boundary condition in equation no. (4), i.e. at

Hence, with the help of equation no (4), we get the un-decayed inventory during

And, the deteriorating items during

Neglecting the higher power of

Total Cost Function: Total average inventory cost per unit time per cycle can be expressed as below,

By using the Equation (3), (6) and (7), we now have,

Now, the objective is to minimize the total inventory cost TC. For the minimum average inventory cost TC the optimum values of time

(i)

and

The cost function will be convex if these well recognized criteria are satisfied. Thereby, we can determine the total optimum cost TC, optimum time interval

Now, using the Equations (10), (11) and (13), we get,

This term will be greater than zero, i.e. convex property (i) will be satisfied, if

putting the values of (9) and (12) in the convex property (i), i.e. in

Again from

Now from the equation no (9) and (10) with the positive value of T from (10), we get,

Solving this equation we now get the value of

Here, we provide a numerical illustration to justify the optimum inventory cost and the optimum order cycle. Let us consider, the inventory system has the following parameters,

Now we study the effects of changes of parameters

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

1)

2)

3)

4)

5)

In a real market, it is very unlikely that the rate of demand always remains same. At times, demand goes very high and at times it goes very low. These types of cases fit the exponential rate of demand. This proposed model expects exponential declining demand. We have developed a time dependent inventory model for the items which have finite shelf-life and time dependent demand with exponential decay. The production rate and the decay are constant all through. Total cost, order interval and total time cycle do not depend on the decreasing

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

+50 | 0.923 | 14.363 | 68.620 | 21.187 | |

+25 | 1.118 | 12.148 | 68.620 | 20.808 | |

+10 | 1.280 | 10.818 | 68.620 | 20.580 | |

−10 | 1.588 | 9.046 | 68.620 | 20.276 | |

−25 | 1.937 | 7.717 | 68.620 | 20.049 | |

−50 | 3.066 | 5.501 | 68.620 | 19.669 | |

+50 | 1.766 | 6.978 | 102.930 | 24.849 | |

+25 | 1.597 | 8.160 | 85.774 | 22.638 | |

+10 | 1.491 | 9.126 | 75.481 | 21.312 | |

−10 | 1.342 | 10.917 | 61.757 | 19.544 | |

−25 | 1.222 | 12.886 | 51.465 | 18.218 | |

−50 | 0.996 | 18.794 | 34.310 | 16.007 | |

μ | +50 | 1.472 | 9.277 | 241.587 | 21.158 |

+25 | 1.445 | 9.592 | 126.405 | 20.793 | |

+10 | 1.429 | 9.793 | 87.184 | 20.574 | |

−10 | 1.406 | 10.077 | 54.397 | 20.282 | |

−25 | 1.389 | 10.302 | 38.958 | 20.063 | |

−50 | 1.360 | 10.706 | 23.308 | 19.698 | |

γ | +50 | 1.418 | 9.932 | 47.872 | 20.428 |

+25 | 1.418 | 9.932 | 57.231 | 20.428 | |

+10 | 1.418 | 9.932 | 63.793 | 20.428 | |

−10 | 1.418 | 9.932 | 73.845 | 20.428 | |

−25 | 1.418 | 9.932 | 82.506 | 20.428 | |

−50 | 1.418 | 9.932 | 99.468 | 20.428 | |

H | +50 | 1.102 | 10.433 | 68.620 | 24.878 |

+25 | 1.234 | 10.226 | 68.620 | 22.653 | |

+10 | 1.336 | 10.063 | 68.620 | 21.318 | |

−10 | 1.514 | 9.777 | 68.620 | 19.538 | |

−25 | 1.699 | 9.484 | 68.620 | 18.203 | |

−50 | 2.208 | 8.716 | 68.620 | 15.978 | |

η | +50 | 1.472 | 9.272 | 68.620 | 21.158 |

+25 | 1.445 | 9.592 | 68.620 | 20.793 | |

+10 | 1.429 | 9.793 | 68.620 | 20.574 | |

−10 | 1.406 | 10.077 | 68.620 | 20.282 | |

−25 | 1.389 | 10.302 | 68.620 | 20.063 | |

−50 | 1.360 | 10.706 | 68.620 | 19.698 |

rate of demand as the rate is considered very small. The model develops an algorithm to determine the optimal demand, optimal order interval, optimal time cycle and the optimum total cost. In this model, the initial demand

Mohammad EkramolIslam,Shirajul IslamUkil,Md. SharifUddin, (2016) A Time Dependent Inventory Model for Exponential Demand Rate with Constant Production Where Shelf-Life of the Product Is Finite. Open Journal of Applied Sciences,06,38-48. doi: 10.4236/ojapps.2016.61005

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