AJORAmerican Journal of Operations Research2160-8830Scientific Research Publishing10.4236/ajor.2016.61008AJOR-63034ArticlesPhysics&Mathematics A Production Inventory Model of Constant Production Rate and Demand of Level Dependent Linear Trend hirajulIslam Ukil1*Md.Sharif Uddin1Jahangirnagar University, Savar, Bangladesh* E-mail:shirajukil@yahoo.com(HIU);11012016060161702 December 2015accepted 22 January 26 January 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Production Inventory Level Dependent Linear Trend Constant Production Rate
1. Introduction

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.

2. Literature Review

3. Assumptions and Notations

・ Production rate is constant and greater than demand rate at any time.

= Demand rate at any instant, where “” and satisfying the condition.

・ For unit inventory, amount of decay rate is very small and constant.

・ 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.

= Inventory level at instant.

= Un-decayed inventory at to.

= Un-decayed inventory at to.

= Deteriorating inventory at to.

= Deteriorating inventory at to.

and Q = Inventory level at time, and respectively. Here, Q is the buffer stock.

= Vary small portion of instant.

= Set up cost.

= Average holding cost.

= Total average inventory cost in terms of.

= Time when inventory gets maximum level.

= Total time cycle.

= Optimum order quantity.

= Optimum time at maximum inventory.

= Optimum order interval.

= Total average optimum inventory cost.

4. Formulation of the Model

Basing on the demand pattern, the business institution decides the structure of the model (Figure 1). This demand very often changes because of various reasons. In reality, the demand may at times the demand be dependent on the level or the stock on hand in the inventory. To meet this type of situation, this model is developed. The model is suitable for those kinds of products which have finite shelf-life and ultimately causes the products decay due to its limited life. At the beginning, while time, the production starts with Q inventories and the production remains constant for entire production cycle.

The inventory increases at the rate of during to. The market demand is and is the decay of inventories at instant where, is the decay of unit inventory in the mentioned period. With the help of the above criteria we can formulate the differential equations as

Inventory model with linear demand

below:

The general solution of the differential equation is.

Applying the following boundary condition, we get at, Solving these equations, we get,

,

Therefore, (1)

From the other boundary condition, i.e. at, , taking up to first degree of, we get the following equation:

Using the Equation (1) and considering up to second degree of for our convenience, the total undecayed inventory during to we get,

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 during to as there is no production after time. The inventory depletes due to market demand and the deterioration of the items. Similar approach as used before can be applied to get another differential equation which is as follows:

The general solution of the differential equation is defined below:

Applying the boundary condition at, we get,.

By solving we get,.

Therefore, (5)

Substituting another boundary condition, i.e. at, , taking up to first order of, we get the following equation:

Now, using Equation (5) and considering up to the first degree of we get the un-decayed inventory during to as:

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

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

Or, (9)

Considering the value as (10)

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 that minimizes the total average inventory cost for the inventory system we shall adopt the convex property. The total average inventory cost is depicted by the equation no (13). To obtain the optimum time cycle and verify Equation (13) as convex in, we must satisfy the following well established convex property,

(i) and

Now differentiating Equation (13) with respect to we get the following equation,

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

Or,

Or, (15)

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

Again differentiating Equation (14) with respect to, we get,

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

both is positive. Finally, we conclude that total cost function (13) is convex in. Hence, there is an optimal solution in for which the total average inventory cost must be minimal.

5. Numerical Search Procedure

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 units, optimum time units at maximum inventory level, optimum order quantity units and total average optimum inventory cost units respectively. Substituting the values of arbitrarily either bigger or lesser than, we get the inventory cost gradually increased from the minimum inventory cost at optimum level, which is show in Table 1 and Figure 2. The table and figure justify the total average optimum inventory cost. If we analyze the table and figures we can observe that in a particular point total inventory cost is minimal and the order interval is optimum.

6. Sensitivity Test

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

Order Interval ()5.2505.5005.7506.0006.2526.5006.7507.0007.250
Total Cost ()52.6752.4552.3052.2152.1952.2152.2852.3952.54
Time verses total cost Effects of the changes of parameters
ParametersChange in %Value of
+501.7365.06168.80870.696
+251.6855.57069.62561.395
+101.6565.95470.11555.852
−101.6206.59270,76948.562
−251.5957.20071.25943.205
−501.5558.59172.07634.569
+501.2657.241111.39248.121
+251.4166.75490.91849.900
+101.5376.45478.63251.205
−101.7626.05262.25253.265
−252.0145.76549.96754.996
−502.8845.50529.49256.826
+501.6975.43866.34757.338
+251.6675.80768.39554.747
+101.6496.06369.62353.200
−101.6276.45771.26151.192
−251.6116.79772.49049.727
−501.5827.47774.53747.378
+501.7355.07263.89060.491
+251.6845.57767.16656.293
+101.6565.95769.13253.812
−101.6216.58871.75350.602
−251.5957.18673.718748.302
−501.5568.54976.99444.752
+501.3375.10570.44270.284
+251.4655.59270.44261.237
+101.5625.96170.44255.808
−101.7276.59070.44248.570
−251.8917.21970.44243.142
−502.3178.84270.44234.095
+501.6356.21770.36052.469
+251.6366.23870.40952.301
+101.6376.24570.42652.245
−101.6386.25970.45952.133
−251.6396.26670.47552.078
−501.6406.28770.52451.910

on the optimal time at maximum inventory level, optimum length of ordering cycle, optimal order quantity and the total average optimum inventory cost per unit time in the model, will be shown in the following table. If the parameters change the values mentioned in Table 2 by adding and subtracting respectively, we see the effect on the on the optimal time at maximum inventory level, optimum length of ordering cycle, optimal order quantity and the total average optimum inventory cost per unit time. While the change of one parameter take place, the other parameter must remain unchanged. Table 2 shows the effect or the sensitivity.

Table 2 shows that small amount of a particular parameter may affect on the values of and even on great extent. On the basis of the results obtained in Table 2, the following observations can be highlighted:

and decrease while and increase with increase in the value of the parameter. Here is highly sensitive to and moderately sensitive to other values.

and increase while and decrease with increase in the value of the parameter and b. Here, and all are moderately sensitive to the values of and.

and decrease, while increases with increase in the value of the parameter. Here, is moderately sensitive to all the values of and.

and decrease and increases, while remain unchanged with increase in the value of the parameter h. Here, h is highly sensitive to the value of and moderately sensitive to all other values.

7. Conclusion

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 total average optimum inventory cost units.

Acknowledgements

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.

Cite this paper

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

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