^{1}

^{2}

^{3}

The present paper focuses an optimal policy of an inventory model for deteriorating items with generalized demand rate and deterioration rate. Shortages are allowed and partially backlogged. The salvage value is included into deteriorated units. The main objective of the model is to minimize the total cost by optimizing the value of the shortage point, cycle length and order quantity. A numerical example is carried out to illustrate the model and sensitivity analyses of major parameters are discussed.

In the recent three decades, rigorous researches have come to existence on inventory models for deteriorating items. Most of the physical goods deteriorate over time. Food items, fruits, vegetables suffer from depletion by direct spoilage while stored. Highly volatile liquids such as alcohol, gasoline and turpentine undergo physical depletion over time through the process of evaporation. Electronic goods, grains, photographic films and radioactive substances deteriorate through a gradual loss of potential or utility with the passage of time. So decay or deterioration of physical items in stock is a very realistic feature and inventory researchers felt the necessary to use this factor into consideration. Generally, deterioration is defined as the natural process that occurs in most of physical items those lose their characteristic over time. It violates the assumption that goods can be held infinitely for future demand. The mathematical modeling on inventory control was started with the work of Harris [

Besides demand and deterioration rate, other factors like allowing shortages are important for modeling of inventory. Shortages usually occur in two cases when the shortage items are totally backlogged and the other case when the items are partially backlogged. In the former case, the customers are not totally willing to accept the items while the latter case customers are only willing to accept the items which can be supplied by the whole sellers in the next period. Various types of inventory models with completely backlogging were discussed by Murdeshwar [

But, in real life situation, during the shortage period, the willingness of a customer to wait for items declines with the length of the waiting time. Backlogging happens due to the lack of raw materials or work in progress or the demand is uncertain. Chang and Dye [

In the classical EOQ models, the demand rate of an item was assumed as constant. However, in the real market situations, the demand rate of any item always acts as a dynamic state. In this context, Silver and Meal [

Other category of inventory models was developed by considering the deterioration rate as the key factor. Ghare and Schrader [

In real market situations, the sellers offer a reduced unit cost called the salvage value of the deteriorated items to the customers to motivate to buy the deteriorated units. In this context, Jaggi and Aggarwal [

In this study, an effort has been made to determine an optimal policy for deteriorating items considering quadratic demand, three parameter Weibull distribution deterioration rate and salvage value. Shortages are permitted to occur and partially backlogged. Among the different patterns of time varying demands, the most realistic approach is to consider the quadratic demand pattern because it represents both accelerated and retarded growth in demand. Quadratic demand is generally represented by

The following assumptions are taken in developing the model.

1) A single product is considered.

2) Replenishment is instantaneous.

3) The lead time is zero.

4) The demand rate is deterministic and quadratic function of time.

5) The deterioration rate is three-parameter Weibull distribution deterioration.

6) The shortages are permitted and backlogged. It is assumed that the backlogging rate will be smaller when the waiting time is longer.

7) During the planning horizon, there is no need to replace or repair the deteriorated units.

8) The salvage value of the deteriorated units depends on the cost deterioration during the cycle time.

The following notations are taken in developing the model.

1)

2)

3)

4)

5)

6)

rameter.

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

17)

18)

19)

The inventory system goes as follow: at time

Using the value of

cation parameter respectively and

Equation (1) is a linear differential equation. The integrating factor (I.F.) is

The solution of Equation (1) with boundary condition

(by neglecting the higher power of

The maximum positive inventory level for each cycle can be obtained by putting

At time

the rate

the differential equation

Using the value of

above equation is given by

The solution of Equation (4) with boundary condition

The maximum back order units are given by

Hence, the order size during the time interval

Now, the total relevant cost of the model is expressed as the difference of the sum of the cost of ordering, cost of carrying inventory, cost of deterioration, cost of shortage due to backlogging and cost of opportunity due to lost sales and salvage value of the deteriorated items.

Now, the per order cost of ordering cost is

The cost of carrying inventory is

(by neglecting the higher power of

The cost of deterioration is

(by neglecting the higher power of

The cost of shortage due to backlogging

The cost of opportunity due to lost sales

The salvage value of deteriorated items per unit time

Thus, from the above arguments, the total annual cost per unit time for the retailer is

The objective of the model is to minimize the total relevant cost per unit time

conditions for minimizing the total relevant cost per unit time are

Equation (15) implies

and

The solutions of (16) and (17) will give the optimal shortage point

and

at

If the solutions obtained from (16) and (17) do not satisfy the sufficient conditions (18), (19) and (20), the optimal solution is infeasible. In that case, either the values of parameters are consistent or there is some error in their estimations.

After obtaining the optimal values of

Example 1. Let us consider the following parametric values of the inventory system as:

Solving the simultaneous Equations (16) and (17), the optimal shortage period and optimal cycle length are obtained as

We study the effects of changes in the parameters of the model such as

Parameter | Change in parameter | % Change in | ||||
---|---|---|---|---|---|---|

+50 +25 −25 −50 | 0.166023 0.151813 0.118055 0.0966262 | 0.221828 0.202687 0.157334 0.12863 | 268.222 244.906 189.797 155.015 | 3229.52 2946.85 2280.32 1860.71 | +0.225862 +0.118566 −0.134436 −0.293711 | |

+50 +25 −25 −50 | 0.111726 0.122117 0.156034 0.188205 | 0.148849 0.162782 0.208371 0.251768 | 268.567 245.092 189.562 154.426 | 3219.38 2941.3 2287.82 1880.56 | +0.222013 +0.116459 −0.131589 −0.286143 | |

+50 +25 −25 −50 | ||||||

+50 +25 −25 −50 | 0.135974 0.136005 | 0.181387 0.181429 | 219.06 219.082 | 2634.9 2634.7 | +0.000155628 +0.0000797118 | |

+50 +25 −25 −50 | 0.136379 0.136208 0.135866 0.135695 | 0.181733 0.181602 0.18134 0.181209 | 219.418 219.261 218.945 218.788 | 2630.08 2632.28 2636.69 2638.89 | −0.00167395 −0.000838872 +0.000835076 +0.00167015 | |

+50 +25 −25 −50 | 0.135028 Complex no. Complex no. 0.134304 | 0.180706 Complex no. Complex no. 0.18012 | 218.183 | 2647.96 | +0.00511294 | |

+50 +25 −25 −50 | 0.136467 0.136251 0.135822 0.135609 | 0.181811 0.181641 0.181301 0.181133 | 219.513 219.308 218.898 218.695 | 2629.51 2632.0 2636.97 2639.45 | −0.00189031 −0.000945154 +0.000941359 +0.00188272 | |

+50 +25 −25 −50 | 0.104755 0.11807 0.162146 0.205308 | 0.157596 0.167553 0.202547 0.239134 | 189.677 201.953 245.073 290.215 | 3041.32 2857.33 2355.91 1989.44 | +0.154425 +0.0845856 −0.105743 −0.244848 | |

+50 +25 −25 −50 | 0.136375 0.136206 0.135868 0.135699 | 0.18173 0.1816 0.181341 0.181213 | 219.422 219.262 218.943 218.785 | 2630.1 2632.3 2636.68 2638.86 | −0.00166636 −0.00083128 +0.00083128 +0.00165877 | |

+50 +25 −25 −50 | 0.140474 0.13852 0.132776 0.128302 | 0.175595 0.178133 0.186057 0.192758 | 212.21 215.195 224.442 232.186 | 2720.32 2682.51 2571.47 2485.05 | +0.0325794 +0.0182274 −0.0239211 −0.0567245 | |

+50 +25 −25 −50 | 0.138628 0.137418 0.134443 0.132586 | 0.17799 0.179597 0.183682 0.186332 | 215.027 216.911 221.681 224.761 | 2684.61 2661.21 2603.69 2567.8 | +0.0190246 +0.0101424 −0.0116911 −0.0253142 | |

+50 +25 −25 −50 | 0.136036 0.136036 0.136036 0.136037 | 0.18147 0.181471 0.181471 0.181471 | 219.102 219.103 219.103 219.103 | 2634.49 2634.49 2634.49 2634.48 | 0 0 0 | |

+50 +25 −25 −50 | 0.138506 0.137349 0.134532 0.132791 | 0.178301 0.179775 0.183443 0.185768 | 215.117 216.963 221.612 224.597 | 2682.29 2659.9 2605.38 2571.68 | +0.0181439 +0.00964513 −0.0110496 −0.0238414 |

Here “

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

In the present paper, an optimal policy for deteriorating items is derived considering quadratic demand rate, a three-parameter Weibull distribution deterioration rate and salvage value. Shortages are permitted and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. Quadratic demand is appropriate for the seasonal fashion items, cosmetic and high-tech products. As deterioration rate starts after some time when the items are stocked. Therefore, a three-parameter Weibull distribution deterioration rate is considered for developing the model. For selling the deteriorated units, salvage value is required for the determination of optimal total cost. Finally, optimal order quantity per cycle and optimal total relevant cost is derived. Shortages are not permitted and partially backlogged. As the rate of deterioration of most items increases with time or age, i.e., the longer the item remains unused, the higher would be its failure rate. Moreover, the location parameter illustrates the shelf-life of the item in the stock. Therefore, the three-parameter Weibull distribution deterioration is suitable for items with any initial value of the rate of deterioration and for items, which start deteriorating only after a certain period of time.

The proposed model can be extended in numerous ways. Firstly, we may extend demand rate to stock dependent demand rate. Secondly, it may be extended to stochastic demand pattern. Finally, we could also extend the model by incorporating quantity discounts, inflation, a finite rate of replenishment and permissible delay in payments etc.

Pandit Jagatananda Mishra,Trailokyanath Singh,Hadibandhu Pattanayak, (2016) An Optimal Policy with Quadratic Demand, Three-Parameter Weibull Distribution Deterioration Rate, Shortages and Salvage Value. American Journal of Computational Mathematics,06,200-211. doi: 10.4236/ajcm.2016.63021