Jon Juneau |
Eyler R. Coates |

School of Engineering Technology |
School of Engineering Technology |

The University of Southern Mississippi |
The University of Southern Mississippi |

Box 5137, Hattiesburg, MS 39406-5137 |
Box 5137, Hattiesburg, MS 39406-5137 |

Jon.Juneau@usm.edu |
Eyler.Coates@usm.edu |

**Abstract**

To manage inventory, the classical economic order quantity (EOQ) equation can be used to decide how much to order. The EOQ is the order quantity that theoretically minimizes the total of the cost of ordering and holding inventory and it assumes that the demand is constant, or does not vary over time. However, when the demand for a product is entering a growth phase, the demand does vary over time. In this paper, a generalized relationship between the demand and the ratio of the ordering cost and the inventory holding cost is derived when the total of the ordering and holding costs is minimized. This generalized relationship holds for the classical EOQ, when the demand is constant, and for demands that vary over time. To demonstrate the generality of the relationship, a procedure is developed for computing the optimal order quantity under conditions of exponentially increasing demand.

**Introduction**

Inventories serve a number of important functions such as meeting anticipated demand, smoothing production requirements, taking advantage of quantity discounts, minimizing the effects of production and delivery disruptions, and hedging against price increases. However, inventories cost money to obtain and keep around. Therefore, two simultaneous pursuits of inventory control are to provide the right material at the right time and to minimize the cost of providing that service.

This paper is concerned primarily with independent demand items, which are generally finished goods. Ordering a large quantity reduces the ordering costs (sometimes called the setup cost) since orders are less frequent. Also, setup costs divided over the units produced and shipping expenses per unit are often reduced. However, ordering a small quantity will reduce the holding cost (due to the tied-up capital in the items) and the storage space required since there will be less inventory on hand. Economic order quantity (EOQ) models have been developed that balance these different costs to obtain a minimum total cost.

There are several assumptions used in the derivation of the economic order quantity:

- knowing the ordering cost and the cost of holding inventory
- instant replenishment of inventory (entire shipment comes in at one time)
- the item is not allowed to experience shortages (at least in the simple EOQ relationship)
- expressions for the ordering cost and the holding cost as a function of the order quantity are required
- the average inventory level with constant demand and instantaneous replenishment will be one-half the order quantity
- the holding cost is assumed to be directly proportional to the average inventory level
- the ordering cost is assumed to be constant for each order
- the demand rate is level and constant from one time period to the next
- the number of orders per year will be the annual demand divided by the order quantity.

The notation in this paper uses common symbols:

C_{h} |
= holding cost (i.e., $ per unit per year) |

C_{O} |
= ordering cost (i.e., $ per order or $ per setup) |

D |
= demand rate (i.e., units per year) |

EOQ |
= economic order quantity (constant demand) |

Q |
= order quantity |

The result is that the total cost will be the sum of two functions: one function (holding cost) that is directly proportional to the order quantity and another function (ordering cost) that is inversely proportional to the order quantity.

Total Inventory Cost = Holding Cost and Ordering Cost = (C * _{h}Q/2) + (C * _{O}D/Q) |
(1) |

It is widely known that by using calculus, it can be shown that the minimum total cost will occur when the two functions, the directly proportional and the inversely proportional functions, are equal to each other. The resulting equation for the economic order quantity is:

(2) |

Most of the earlier research on setup cost reduction with EOQ models assumes that demand and lead time are constant but recent research has challenged these assumptions^{1, 2, 5, 6, 8, 10, 11, 12, 13}. In regard to relaxing the constant demand assumption, there are two general approaches. One approach relaxes the constant demand assumption by assuming that the demand is either stochastic^{2, 5, 6} or is lumpy due to batching and pull production systems^{11}. Another approach relaxes the constant demand assumption by assuming that the underlying rate of demand is changing^{1, 12, 13}.

This paper is concerned with the second approach (i.e., the underlying rate of demand is changing). Chen (1998) and Zhao *et al* (2001) discuss EOQ approaches when the rate is changing linearly. Yang *et al* (1999) look at heuristics that find the EOQ when the demand is increasing in a non-linear fashion. In contrast to earlier research, an analytical approach will be developed to calculate the EOQ when the demand follows any trend, linear or non-linear, that can be expressed analytically. To demonstrate the generality of this approach, the relationship was used to develop the EOQ when the demand behaves in an exponential manner.

Exponential demand in an EOQ equation is being developed because of the potential for wide use among businesses, especially new businesses and industries. Hamblin *et al.* (1973) have stated, "Exponential growth often holds over certain periods, or epochs."^{3} Porter (1991) states, "The growth rate then sifts and another epoch emerges. Over multiple epochs, continuing exponential growth becomes apparent, but succeeds at different rates. Exponential growth may hold for the time period of interest, but possible physical or social limits that could slow or stop growth must be anticipated." ^{9} Also, Porter (1991) mentions that "Exponential behaviors often occur at the beginning of growth regimes. Many exponential growth patterns will in fact begin to level out and produce curves more characteristics of the Pearl or Gompertz models, when observed for a sufficient length of time. The Lotka-Volterra equations will produce solutions that approximate the exponential models when the market is not saturated. It should be noted again that the Lotka-Volterra equations will not produce a closed-form exponential curve. They will, however, produce a curve that is equivalent to the exponential model for any arbitrary defined level of accuracy and any desired length of time."^{9}

There are a number of examples in the literature where analysts were confronted with exponential growth situations. Long (1999) describes the exponential growth of undersea fiber-optic cable systems to support the internet age^{4}. Also, Mutooni and Tennenhouse (2000) use exponential growth to model the growth of communication network capacities. They also point out that exponential growth is very common for hi-tech industries^{7}. Since there are many new industries that are experiencing rapid growth, generating EOQ equations that can handle time varying demand has become an important priority. Thus, the present investigation is warranted.

The paper will be organized as follows: In the first main section, a relationship will be derived that minimizes the total ordering and holding costs for an inventory item that has a demand trend. The EOQ model with this type of varying demand is designated as EOQ-vD. Then in the second main section, constant demand is assumed with the new model to demonstrate that the new model reduces back to the classical EOQ model. In the third section, the relationship derived in the first section will be used to develop a procedure that can be used when the demand behaves in an exponential manner. In the fourth main section, the procedure will then be demonstrated with an example application. Finally, some recommendations for further enhancements will be presented.

EOQ for Generalized Demand, EOQ-vD

In this section, the classical EOQ will be extended by allowing the demand to vary over time. To begin, consider that the inventory is dependent on time and can be represented by a differential equation.

The differential equation that describes the behavior of the inventory over time where *D(t)* is the demand at time, *t* is given by:

(3) |

The boundary conditions for equation (3) are: *I(T)*=0 and *I(t _{0})*=

Solving equation (3) for the inventory over time, *I(t)*, yields:

(4) |

From equation (4), one can also obtain the order quantity, *Q*, because the inventory at the beginning of the order cycle is the quantity ordered.

(5) |

The average inventory has to be determined in order to compute the holding cost. The average inventory is given by the integration of the inventory level over the order cycle divided by the time of the order cycle.

(6) |

Changing the order of integration yields:

(7) |

Integrating results in the following expression for the average inventory, as a function of the time dependent demand:

(8) |

The derivation becomes easier when the costs are expressed as a function of the duration of the order cycle. The ordering cost occurs once over an order cycle. Thus the total of the ordering cost and the holding cost becomes:

(9) |

Substituting equation (8) into (9) yields:

(10) |

The minimum costs as a function of the cycle duration can be found by taking the derivative of the total cost equation (10) and equating the derivative to zero.

(11) | |

(12) |

Using Liebnitz’s rule to take the derivative of the integral results in the following:

(13) |

Substituting the results of (13) into (12) gives:

(14) |

Multiplying by (*T-t _{0}*) and combining terms results in the following:

(15) |

Then, using the substitution *x* = *T – u*, results in the final relationship:

(16) |

The classical EOQ Re-derived

In this section, the general relationship given in equation (16) is shown to apply to the classical EOQ. Showing how the relationship can be used to derive the classical EOQ provides guidance into how the relationship can be used for other time behaviors of demand. In the classical EOQ, the demand is a constant. Let the demand, *D(t)* = *D*_{0}. Then the relationship in (16) can be restated as:

(17) |

The EOQ, *Q*, must also be known. This is possible using equation (5):

(18) |

By multiplying (17) by *D*_{0}, and substituting the expression for *Q* from (18) results in the following:

(19) |

Solving (19) for *Q*, gives the classical EOQ formula:

(20) |

EOQ for Exponential Demand

The first step to use the optimum relationship in (16) to determine the economic order quantity is to determine the equation for the demand over time. For an exponential demand, let *D*(*t*) = *D*_{0}*e ^{a
t}* where a
is a parameter that represents the exponential growth in demand.

The order quantity can be expressed by applying (5) to obtain:

(21) |

If the cycle length, (*T*-*t*_{0}) were known, then the calculation of the order quantity, Q could be given by (21). We can use the optimal relationship in (16) to obtain the optimal cycle length.

For exponential growth in demand, the optimum relationship in (16) becomes:

(22) |

Evaluating the integral in (22) leads to:

(23) |

Substituting and then dividing
by *D*(*t*_{0}) results in:

(24) |

Defining the value of simplifies (24) into:

(25) |

The quantity on the left side of (25) is a constant and can be redefined as:

(26) |

The last step to determine the optimal cycle time is to obtain a value for b that satisfies (25). The recommended procedure to do this is to use the Newton’s method. The initial guess for b can be derived by expanding the generalized EOQ relationship in a Taylor series in b and then ignoring the terms higher than the second power of b . Use the following as an initial value for b :

(27) |

Then use the following equation (28) for the recursion:

(28) |

Once the value of b is determined, recall the substitution, and find the length of the next order cycle by:

(29) |

Equation (21) can now be used directly to obtain the optimal quantity:

(30) |

An Example Application and Comparison to Classical EOQ Models

An example of how to use this economic order quantity is presented in this section. To determine some typical ranges of exponential growth, data from the annual reports of several publicly traded companies were obtained. To represent the use of inventory for the companies, an exponential curve was fit to the annual cost of goods sold. A large established company, General Electric Company, does not show much growth with an estimate of a equal to 0.085. A newer company with fast growth in the cost of goods sold, Amazon.com, has an estimated a equal to 1.99. Two other companies, WorldCom, Inc. and Cisco Systems, had estimated a values of 0.459275 and 0.440404, respectively. It appears that the exponential growth rates for typical companies will have values of a between zero to about 2 for very fast growing companies, and about 0.45 for other typical growth companies.

Since the cost of ordering and cost of holding inventory for these companies are not known, for this example, we assume that the cost of holding inventory will be about $1.00 per unit per year and the cost of ordering the inventory will be about $25. Also, for this example, the most recent annual cost of goods sold was used as the current demand.

The economic order quantities, for constant demand and for exponentially increasing demand, for several companies are shown in the following table:

General Electric | Cisco Systems | WorldCom, Inc. | Amazon.com | |

Last Annual COGS | 45958 | 6746 | 15951 | 1349.2 |

a | 0.0854 | 0.4404 | 0.4593 | 1.9903 |

Classical EOQ | 1516 | 581 | 893 | 250 |

EOQ_exponential | 1517 | 588 | 901 | 294 |

Order Cycle Time | 0.0330 | 0.0856 | 0.0557 | 0.1810 |

As illustrated with the numbers for General Electric, when the exponential growth rate is small, the economic order quantity for the exponential demand growth is very close to the classical economic order quantity. As expected, since Cisco Systems and WorldCom, Inc. have similar values for a , the difference between the classical and the exponentially increasing demand economic order quantities are both about 1%. For these situations, it appears that using the traditional EOQ will be sufficient for determining the order quantity. When the growth gets large, such as Amazon.com’s a value of 1.9903, the difference between the economic order quantities does become significant.

Concluding Remarks

An EOQ model has been derived that can be used for varying demand rates, EOQ-vD. The generality of the EOQ-vD model has been demonstrated by showing that it can be used to determine the economic quantity when demand is growing exponentially. It was shown that the EOQ-vD model reduces to the standard EOQ model when the demand rate is set to constant.

For further research in this area, the model can be extended to include finite replenishment rates so that the economic manufacturing quantity can also be determined. Sarker and Coates (1997) have stated that the inclusion of finite replenishment rates may have a larger impact on the order quantity than the recognition of variable lead times.^{10} This recognition of finite replenishment rates may also be as important in EOQ models with variable demand rate. Quantity discounts should also be included in the EOQ-vD model since discounts are a common feature in the business environment.

Because the generalized relationship developed in this paper can be applied to other demand behaviors, the model can be used in a large variety of situations. The extension of the EOQ model to include varying demand should be of particular interest to the practitioner since most companies are not in a steady-state business environment and demand is not constant.

References

1. | Chen, J. (1998) "An Inventory Model for Deteriorating Items with Time-proportional Demand and Shortages under Inflation and Time Discounting, " |

2. | Gavirneni, S. and Taylor, S. (2001) "An Efficient Procedure for Nonstationary Inventory Control, " |

3. | Hamblin, R.L., Jacobsen, R. B., and Miller, J.L. (1973) A Mathematical Theory of Social Change (New York, NY: John Wiley & Sons, Inc.) |

4. | Long, S.H. (1999) "Forecasting Fiber Optic Undersea Cable Systems in the Internet Age, " |

5. | Matheus, P. and Gelders, L. (2000) "The (R, Q) Inventory Policy Subject to a Compund Poisson Demand Pattern, " |

6. | Moinzadeh, K. (2001) "An Improved Ordering Policy for Continuous Review Inventory Systems with Arbitrary Inter-Demand Time Distributions," |

7. | Mutooni, P. and Tennenhouse, D. (2000) "Modeling the Communication Networks' Transition to a Data-Centric Model, " http://www.ksg.harvard.edu/iip/Dean/hiip/Activities/Conferences/iicp/papers/mutooni.htm, Revised: August 30, 2000 |

8. | Nasri, F., Affisco, J.F., and Paknejad, M.J. (1990) "Setup Cost Reduction in an Inventory Model with Finite-Range Stochastic Lead Times," |

9. | Porter, A.L., Roper, A.T., Mason, T.W., Rossini, F.A., and Banks, J. (1991) Forecasting and Management of Technology (New York, NY: John Wiley & Sons, Inc.), 191- 92 |

10. | Sarker, B.R. and Coates, E.R. (1997), "Manufacturing setup cost reduction under variable lead times and finite opportunities for investment," |

11. | Sarker, B.R. and Parija, G.R. (1996), "Optimal Batch Size and Raw Material Ordering Policy for a Production System with a Fixed Interval, Lumpy-Demand, Delivery System," |

12. | Yang, J., Zhao, G.Q. and Rand, G.K. (1999), "Comparison of Several Heuristics Using An Analytic Procedure for Replenishment with Non-Linear Increasing Demand," |

13. | Zhao, G.Q., Yang, J., and Rand, G.K. (2001), "Heuristics for Replenishment with Linear Decreasing Demand," |