Year:
2011

Volume:
6

Issue:
3

Page No.
46 - 53

Authors : Gour Chandra Mahata

**Abstract:** In this study, an order-level inventory model is constructed
for deteriorating items with instantaneous replenishment, exponential decay
rate and a time varying linear demand without shortages under per-missible delay
in payments. It is assumed that a constant fraction of the on-hand inventory
deteriorates per unit of time. The exact formulae of the optimal average cost
and the lot size are derived without carrying out any approximation over the
deterioration rate. Different decision making situations illustrated with the
help of numerical examples. Sensitivity analysis of the optimal solution with
re-spect to changes in the parameter values is carried out.

Gour Chandra Mahata , 2011. EOQ Model for Items with Exponential Distribution Deterioration and Linear Trend Demand under Permissible Delay in Payments. *International Journal of Soft Computing, 6: 46-53.*

**INTRODUCTION**

In formulating inventory models, two factors of the problem have been of growing interest to the researchers, one being the deterioration of items and the other being the variation in the demand rate.

Demand is the major factor in the inventory management. Therefore, decisions
of inventory are to be made because of the present and future demands. As demand
plays a key role in modeling of deteriorating inventory, researchers have recognized
and studied the variations (or their combinations) of demand from the viewpoint
of real life situations. Demand may be constant, time-varying, stock-dependent
and price-dependent, etc. The constant demand is valid only when the phase of
the product life cycle is matured and also for finite periods of time. Wagner
and Whithin (1958) discussed the discrete case of the dynamic version of
EOQ. Covert and Philip (1973), Misra
(1975), Dave (1979) and Sarma
(1987), etc., established inventory models with constant demand rate. The
classical no-shortage inventory policy for linear trend in demand was discussed
by Donaldson (1977). Ritchie (1980,
1984) discussed the solution of the EOQ formula for
a linear increasing time-dependent demand which was obtained by Donaldson
(1984). EOQ models for deteriorating items with trended demand were considered
by Bahari-Kasani (1989), Goswami
and Chaudhuri (1991, 1992), Xu
and Wang (1990), Chung and Ting (1993, 1994),
Kim (1995), Jalan *et al*. (1996),
Jalan and Chaudhuri (1999) and Lin
*et al*. (2000), etc. In this extent, Begum *et
al*. (2009) developed an inventory model with exponential demand rate,
finite production rate and shortages. Silver and Meal (1969)
developed an approximate solution procedure for the general case of time-varying
demand. Generally, this type of demand exists for some particular goods. Many
research articles by Silver (1979), Henery
(1979), McDonald (1979), Dave
and Patel (1981), Sachan (1984), Deb
and Chaudhuri (1986), Murdeshwar (1988) and Hargia
(1993), etc., analyzed linear timevarying demand. Later, Ghosh
and Chaudhuri (2004, 2006), Khanra
and Chaudhuri (2003) and Begum *et al*. (2010),
etc., established their models with quadratic time-varying demand. Teng
and Chang (2005) established an economic production quantity models for
deteriorating items with demand depend on price and stock. The deterioration
rate of inventory in stock during the storage period constitutes an important
factor which has attracted the attention of researchers. In inventory problems,
deterioration is defined as damage, decay, spoilage, evaporation, obsolescence
and loss of utility or loss of marginal value of goods that results in decrease
the usefulness of the original one. Whitin (1957) is the
first reseacher who studied an inventory model for fashion goods deteriorating
at the end of a prescribed storage period. An exponentially decaying inventory
was developed by Ghare and Schrader (1963). Emmons
(1968) established a replenishment model for radioactive nuclide generators.

The assumption of the constant deterioration rate was relaxed by Covert and Philip (1973) who used a two parameter Weibull distribution to represent the distribution of time to deterioration. This model was further generalized by Philip (1974) by taking three parameter Weibull distribution deterioration. Shah and Jaiswal (1977) established an order-level-inventory model for perishable items with a constant rate of deterioration.

Recently, Begum *et al*. (2010) has discussed
an EOQ model for the deteriorating items with two parameter Weibull distribution
deterioration.

The problem of determining the Economic Order Quantity (EOQ) under the condition of a permissible delay in payment has drawn the attention of researchers in recent times. It is assumed that the supplier (wholesaler) allows a delay of a fixed period for settling the amount owed to him/her. There is no interest charged on the outstanding amount if it is paid within the permissible delay period.

Beyond this period, interest is charged. During this fixed period of permissible delay in payments, the customer (a retailer) can sell the items, invest the revenues in an interest-earning account and earn interest instead of paying off the over-draft which is necessary if the supplier requires settlement of the account immediately after replenishment. The customer finds it economically beneficial to delay the settlement to the least moment of the permissible period of delay. This problem was first studied by Goyal (1985) for a non-deteriorating items having a constant demand rate. Chand and Ward (1987) commented in a brief note on some of the assumptions made by Goyal (1985) in analyzing the cost of funds tied up in inventory. The effects of deterioration of goods in stock on the cost and price components cannot be ignored in practice. The model of Goyal (1985) was extended by Aggarwal and Jaggi (1995) to the case of a deteriorating item. Hwang and Shinn (1997) discussed lot sizing policy for an exponentially deteriorating products under the condition of permissible delay in payments when the demand rate would depend on retail price.

In the present study, the researchers assume that the time dependence of demand is linear. Deterioration rate is assumed to be exponential distribution. The rate of replenishment is infinite and shortages are not allowed. The results presented in this study extend and improve the corresponding results of Aggarwal and Jaggi (1995) by taking into account a time-dependent demand rate. The solution procedure involving different decission making situations is illustraded with the help of numerical examples. Analysis is carried out to study sensitivity of the optimal solution to changes in the values of the different parameters involved in the system.

**NOTATION AND ASSUMPTIONS**

The mathematical model is developed on the basis of the assumptions and notations.

**Notation:**

R (t) | = | Demand rate which is linearly dependent on time i.e., R(t) = a+bt where a>0, b>0 |

θ | = | Constant rate of deterioration |

A | = | Ordering cost per order |

c | = | Unit purchasing cost per item |

s | = | Unit selling price per item |

I_{e} |
= | Interest earned per $ per year |

I_{c} |
= | Interest charged per $ in stocks per year |

M | = | Permissible period (in years) of delay in settling the accounts with the supplier |

T | = | Time interval (in years) between two successive orders |

**Assumptions:**

• | The demand rate for the item is represented by a linear and continuous function |

• | Replenishment rate is infinite and replenishment is instantaneous |

• | The lead time is zero |

• | Shortages are not allowed |

• | The distribution of time to deterioration of an item follows the exponential distribution g (t): |

Where, θ is called the deterioration rate; a constant fraction θ, assumed to be small of the on-hand inventory gets deteriorated per unit time during the cycle time.

• | There is no repair or replenishment of deteriorated units in the given cycle |

• | s≥c, I_{c}≥I_{e} |

• | When T≥M, the account is settled at time T = M and retailer starts
paying for the interest charges on the items in stock with rate I_{c}.
When T≤M, the account is settled at T = M and the retailer does not need
to pay interest charge |

• | The retailer can accumulate revenue and earn interest after his/her customer
pays for the amount of purchasing cost to the retailer until the end of
the trade credit period offered by the supplier. That is the retailer can
accumulate revenue and earn interest during the period N to M with rate
I_{e} under the condition of trade credit |

**MATHEMATICAL FORMULATION**

(1) |

Where:

(2) |

And:

(3) |

Where, a>0, b>0. The solution of Eq. 1 using Eq. 2 and 3 is:

(4) |

The initial order quantity at t = 0 is:

(5) |

The total demand during one cycle is:

Number of deteriorated units:

(6) |

The cost of stock holding for one cycle is:

Hence, the holding cost per unit time is:

(7) |

**Case 1
**

Hence, interest payable per unit time is:

(8) |

Interest earned per unit time is:

(9) |

Total variable

Hence, total variable cost per unit time in this case is given by:

(10) |

The researchers have now to minimize C_{1}(T) for a given value of M. The necessary and sufficient conditions to minimize C_{1} (T) for a given value of M are respectively:

(11) |

And:

(12) |

After simplification:

yields the following nonlinear equation in T:

(13) |

By solving Eq. 13 for T, we obtain the optimal cycle length
T = T_{1}* provided it satisfies Eq. 12. The EOQ
q_{0}* for this case is given by:

The minimum annual variable cost C_{1}(T_{1}*) is then obtain from Eq. 10 for T = T_{1}*.

**Case 2
**

and interest earned during (M–T) i.e., up to the permissible delay period is:

Hence, the total interest earned during the cycle is:

(14) |

Hence, the total variable cost per unit time is:

(15) |

The researchers have now to minimize C_{2}(T) as before for a given value of M. After simplification:

yields the result:

(16) |

The optimal cycle length T = T_{2}* which minimizes C_{2}(T) is obtained by solving Eq. 16 for T by using the Newton-Raphson method, provided:

The EOQ in this case is given by:

and the minimum annual variable cost C_{2}(T_{2}* ) is obtained from Eq. 15 for T = T_{2}*.

**Case 3
**

(17) |

The EOQ is:

Now in order to obtain the economic operating policy, the following steps are to be followed:

• | Step 1: Determine T_{1}* from Eq. 13.
If T_{1}*≥M, obtain C_{1} (T_{1}* ) from Eq.
10 |

• | Step 2: Determine T_{2}* from Eq. 16. If T_{2}*<M,
evaluate C_{2}(T_{2}* ) from Eq. 15 |

• | Step 3: If T_{1}*<M and T_{2}*≥M then, evaluate
C(M) from Eq. 17 |

• | Step 4: Compare C_{1}(T_{1}* ), C_{2}(T_{2}*
) and C(M) and take the minimum |

**NUMERICAL EXAMPLES**

The numerical examples given, covers all the three cases that arise in this model:

**Example 1 (Cases 1 and 2):** Let a = 1000 units year^{-1}, b = 150 units year^{-1}, I_{p} = 0.15 year^{-1}, I_{e} = 0.13 per year, s = Rs. 200 per order, h_{p} = Rs. 0.12 year^{-1}, p = Rs. 20 per unit, M = 0.25 year, θ = 0.20.

Solving Eq. 13, we have T_{1}* = 0.284 and the minimum average cost is C_{1}(T_{1}*) = 1283.53. Solving Eq. 16, the researchers have T_{2}* = 0.206 and the minimum average cost is C_{2}(T_{2}*) = 1263.53. Here T_{1}*>M and T_{2}*<M both hold and this implies that both the cases 1 and 2 hold. Now C_{2}(T_{2}*)<C_{1}(T_{1}*). Hence, the minimum average cost in this case is C_{2}(T_{2}*) = Rs. 1263.53 where the optimal cycle length is T_{2}* = 0.206 year <M. The economic order quantity is given by q_{0}* (T_{2}*) = 213.82 units.

**Example 2 (Case 1):** Let a = 1000 units year^{-1}, b = 150 units year^{-1}, I_{p} = 0.15 year^{-1}, I_{e} = 0.13 year^{-1}, s = Rs. 200 per order, h_{p} = Rs. 0.12 year^{-1}, p = Rs. 20 per unit, M = 0.25 year, θ = 0.01.

Solving Eq. 13 for T, the researchers get T_{1}* = 0.432 and the minimum average cost is C_{1}(T_{1}*) = 585.31. Again, solving Eq. 16, the researchers have T_{2}* = 0.274 and the corresponding minimum average cost is C_{2}(T_{2}*) = 793.94.

Here T_{2}*>M which contradicts case 2. In this case T_{1}>M which is case 1. Therefore, the minimum average cost in this case is C_{1}(T_{1}*) = Rs. 585.31, the EOQ is q_{0}* (T_{1}*) = 447.23 units and the optimal cycle length is T_ 1 = 0.432 year >M.

**Example 3 (Case 2):** Let a = 1000 units year^{-1}, b = 150 units year^{-1}, I_{p} = 0.15 year^{-1}, I_{e} = 0.13 year^{-1}, s = Rs. 200 per order, h_{p} = Rs. 0.12 year^{-1}, p = Rs. 40 per unit, M = 0.25 year, θ = 0.20.

Solving Eq. 13 for T, the researchers obtain the optimal value T_{1}* = 0.232 and the optimal cost C_{1} (T_{1}*) = 1792.29. Here, T_{1}*<M which contradicts case 1. Again, solving Eq. 16, the researchers have T_{2}* = 0.147 and the minimum average cost is C_{2} (T_{2}*) = 1395.29. In this case, T_{2}*<M which is case 2. Therefore, the minimum average cost in this case is C_{2} (T_{2}*) = Rs. 1395.29, the EOQ is q_{0}* (T_{2}*) = 150.81 units and the optimal cycle length is T_{2}* = 0.147 year <M.

**Example 4 (Case 3):** Let a = 1300 units year^{-1}, b = 100 units year^{-1}, I_{p} = 0.5 year^{-1}, I_{e} = 0.01 year^{-1}, s = Rs. 97 per order, h_{p} = Rs. 0.12 year^{-1}, p = Rs. 40 per unit, M = 0.09 year, θ = 0.3.

In this case T_{1}* = T_{2}* = 0.09 = M which is case 3. The optimal cost in this case is C (M) = Rs. 2050.56 and the EOQ is q_{0}* = 119.01 units for M = 0.09 year.

Table 1: | Sensitivity of the optimal solution to changes in parameter values |

Table 2: | Results for p = Rs. 20 per unit |

**SENSITIVITY ANALYSIS**

The researchers now study sensitivity of the solution of the problem to changes in the values of the parameters of the systems. Example 3 is used for this purpose and the results are shown in Table 1. It is found that the solution is not sensitive to changes in the values of the parameters b and Ip. However, it is sensitive to changes in the values of the parameters a, h_{p}, I_{e}, p, s, θ and M.

In Table 2, we present the optimal solutions, as M and θ
vary, for a less expensive item (p = Rs. 20) when the parameter values are a
= 1000, b = 150, h_{p} = 0.12, I_{p} = 0.15, I_{e} =
0.13 and s = 200 in appropriate units. We observe the the following characteristics
of the solution:

• | The cycle length increases marginally, the order quantity increases slightly and the cost decreases slightly as the credit period M increases, keeping θ fixed |

• | As the value of θ increases, keeping the credit period M fixed, there is significant reduction in both the cycle length and the order quantity while the cost increases considerably |

Table 3: | Results for p = Rs. 40 per unit |

A similar analysis is made in Table 3 for a more expensive (p = Rs. 40) item. The same type of results is observed in this case also.

Table 4: | Results for p = Rs. 200 per unit |

In Table 4, it is observed that the cycle length, the order quantity and the average system cost all undergo considerable changes for a very expensive (p = Rs. 200) item.

**CONCLUSION**

The present model seeks an extension of Aggarwal and Jaggi (1995) work by taking a time-dependent demand rate into consideration. This consideration makes the model more realistic. A linear, time-dependent demand rate implies a steady increase in the demand of the product. This type of demand pattern is observed in the market in the case of many products. Several numerical illustrations are given to explain the solution procedure of the model. Sensitivity analysis is also carried out. Comparison of this model with that of Aggarwal and Jaggi (1995) shows that there are slight changes in the optimal solutions when a time-dependent demand rate is taken into consideration. As a result of the demand rate varying linearly with time, both the cycle length and the order quantity decreases whereas the average system cost increases in comparison to the results of Aggarwal and Jaggi (1995).

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