**Introduction**

Since manufacturing businesses generally obtain raw materials from different suppliers, the lead time variance of arrival time can cause difficulty in figuring out how frequently they should order raw materials and in what lot sizing to guarantee a smooth and uninterrupted manufacturing operation (Sedarage and Fujiwara, 1997). Economic order quantity (EOQ) model, economic production lot size (EPL) model and the (Q, r) model, play a vital role in inventory control and to find answers to the above mentioned problems in the industry. Economic order quantity model (EOQ) was first modelled by Ford W. Harris in 1913 to make a proper balance between lot size and setup cost (Hopp and Spearman, 2011). One of the main extensions of economic order quantity (EOQ) model is the economic production lot (EPL) model. It focuses on the fact that replenishment is not being instantaneous in the real world, even though demand is constant and known (Hopp and Spearman, 2011). In many cases, the managers have to determine how many socks to carry as in the base stock model and also how much to produce or order at one time as in the economic order quantity (EOQ) model. The (Q, r) model has developed to address both of these cases, including both of these concepts (Hopp and Spearman, 2011).

The purpose of his paper is to compare economic order quantity (EOQ) model, economic production lot (EPL) model and (Q, r) model. In the background section, I have mentioned the basic assumptions of these models and equations that are used to calculate required variables. In the methodology section, I have used two questions from the “Factory Physics” by Hopp and Spearman to evaluate these models. Further, in the results and discussion section, I have discussed the results I acquired from the calculations. At the end of the paper I have mentioned the conclusion after examining these three models.

**Background**

**Economic Order Quantity Model**

Economic order quantity model (EOQ) can be used to determine the total cost of the process and the optimal order quantity. This model is based on 6 major assumptions.

- Production is instantaneous
- Delivery is immediate
- Demand is deterministic
- Demand is constant over time
- A production run sustains a static setup cost
- Products can be analyzed individually

In this model, we can get a cost function and using it, we can compute the optimal production lot size

Y(Q)=hQ/2+AD/Q+cD--------(1)

Where,

h – Holding cost

Q –Lot size

D – Demand rate (units per year)

A – Fixed setup cost to produce a lot

c- Unit production cost

Using cost function, we can formulate the equation to calculate the optimal lot size.

Q=√(2AD/h)-----------(2)

**Economic Production Lot Model**

In the economic production lot model, they assume that production is not instantaneous. Therefore, it develops the equation for calculating the optimal quantity as,

Q=√(2AD/(h(1-D/P))) ______________________(3)

Where, P is the production rate.

When P= ∞, this equation becomes the normal EOQ model. This model consequences in greater lot size to fit for the fact that replenishment times take to produce.

In both EOQ and EPL models, the decision variable is Q which is the lot size depending on the situation.

**(Q, r) Model**

This model focuses on how much stock to carry (base stock model) and how much to produce or order at a time (EOQ and newsvendor model). However, the major assumptions in a (Q, r) model are,

- Continuous review of inventory
- Demands occur one at a time.
- Unfilled demand is backordered.
- Replenishment lead times are fixed and known.

The main decision variables are

- Reorder point (r)
- Order Quantity (Q)

Moreover, the performance measures of a (Q, r) model are

F (Q, r) – average order frequency

F( Q,r)=D/Q _________________(4)

This equation is also called as replenishment orders per year as a function of Q and r.

S (Q, r) – average service level

S(Q,r)=1-1/Q (B(r)-B(r+Q))____________(5)

This equation is also called as fraction orders filled stocks per year as a function of Q and r. When Q =1 , S (1, r)= S(R)= base stock fill rate.

B (Q, r) –average backorder level

B(Q,r)= 1/Q (β(r)-β(r+Q))________________(6)

Where,

β=σ^2/2((z^2+1)(1-φ(z))-zφ(z)) and

z=(x-θ)/σ

σ is the standard deviation.

When Q =1, B (1, r) = B(R)= base stock backorder level.

I (Q, r) – average inventory level

I(Q,r)=(Q+1)/2+r-θ+B(Q,r)________________(9)

When Q =1, I (1, r)= I(R)= base stock inventory level.

**Methodology**

This paper uses 2 questions from the Hopp’s and Speaman’s “Factory Physics.”

**Example 1:**

This example uses the EOQ model to compute optimal order quantity and order quantity.

A gift shop sells little lentils-cuddly animal dolls, stuffed with dried lentils- at a very steady pace of 10 per day, 310 days per year. The wholesale cost of a doll is $5, and the gift shop uses an animal interest rate of 20 percent to compute holding costs.

If the shop wants to place an average of 20 replenishment orders per year, what order quantity should it use?

Using equation (4),

D=10*310= 3100

F= 20

Q= 3100/20 = 155

If the shop estimates the cost of placing an order to be $10, what is the optimal order quantity?

Using equation (2),

A =$10

H= 0.20*5 = 1

D=3100

Q= √((2*10*3100)/1) ~ 248

According the example, the EOQ model is insensitive to variations in demand. Moreover, the optimal order quantity is greater than the order quantity in this case.

**Example 2: **

In the example 2, the paper studies how to do computations using (Q, r) model.

Jill, the office manager of a desktop publishing outfit, stocks replacement toner cartridges for laser printers. Demand for a cartridge is approximately 30 per year and is quite variable. Cartridges cost $100 each and require 3 weeks to obtain from the vendor. Jill uses a (Q, r) approach to control stock levels.

If Jill wants to restrict replenishment orders to twice per year on average. What batch size Q should she use?

Using equation (4)

D=30 per year

F= 2 per year

Q= 30/2 =15

The Q size is 15.

If Jill wants to restrict replenishment orders to 6 times per year on average. What batch size Q should she use?

Using equation (4)

D=30 per year

F= 6 per year

Q= 30/6 =5

The Q size is 5.

Using this batch sizes, what reorder point should she use to ensure service level of 98% ?

Computing mean (θ) when 30 units per year and the lead time is 3 weeks,

θ = (30*3)/52 = 1.730769231

Standard deviation = √θ = 1.315587029

Using equations (5) and (6), below table was constructed to find the reorder point compatible with the service level according to the given order quantities.

Therefore, when Q= 15, the reorder point (r ) is 5 and when Q= 5, the reorder point is ( r ) is 4.

**Results and Discussion**

The economic production lot model is an extension of the economic order quantity model. We use economic production lot model when production is not instantaneous. When production rate is equal to infinity, economic production lot model behaves like the economic order quantity model. Still, in the economic order quantity model, optimal order quantity can be greater or smaller the order quantity. The optimal order quantity depends on setup cost, demand and holding cost.

However, the (Q, r) model implies that increasing demand tends to increase the optimal order quantity; Increasing quantity tends to increase the optimal reorder point; Increasing the variability of the demand process tends to increase the optimal reorder point; Increasing the holding cost tends to decrease the optimal order quantity and reorder point.

**Conclusion**

In summary, the similarities among these three models include that continuous review time of inventory, single product model, deterministic production, setup or other cost, infinite horizon and single echelon production line. EOQ and EPL models are almost the same except for having a finite production rate in EPL model, but (Q, r) model also has an infinite production rate as EOQ model. Other than that, (Q, r) model has backordering and random demand, which both EOQ or EPL models don’t have.

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Works Cited

Hopp, Wallace J. and Mark L. Spearman. *Factory
Physics*. Long Grove: Waveland Press. Inc, 2011. print.

Sedarage, Dayani and Okitsugu Fujiwara. "An
optimal (Q,r) policy for a multi part assembly system under stochastic part
procurement lead times." *European Journal of Operation Research*
(1997): 550-556. Web.