# Economic production quantity

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The economic production quantity model (also known as the EPQ model) determines the quantity a company or retailer should order to minimize the total inventory costs by balancing the inventory holding cost and average fixed ordering cost. The EPQ model was developed by E.W. Taft in 1918. This method is an extension of the economic order quantity model (also known as the EOQ model). The difference between these two methods is that the EPQ model assumes the company will produce its own quantity or the parts are going to be shipped to the company while they are being produced, therefore the orders are available or received in an incremental manner while the products are being produced. While the EOQ model assumes the order quantity arrives complete and immediately after ordering, meaning that the parts are produced by another company and are ready to be shipped when the order is placed.

In inventory management, economic order quantity (EOQ) is the order quantity that minimizes the total holding costs and ordering costs. It is one of the oldest classical production scheduling models. The model was developed by Ford W. Harris in 1913, but R. H. Wilson, a consultant who applied it extensively, and K. Andler are given credit for their in-depth analysis.

## Contents

In some literature, "economic manufacturing quantity" model (EMQ) is used for "economic production quantity" model (EPQ). Similar to the EOQ model, EPQ is a single product lot scheduling method. A multiproduct extension to these models is called product cycling problem.

## Overview

EPQ only applies where the demand for a product is constant over the year and that each new order is delivered/produced incrementally when the inventory reaches zero. There is a fixed cost charged for each order placed, regardless of the number of units ordered. There is also a holding or storage cost for each unit held in storage (sometimes expressed as a percentage of the purchase cost of the item).

We want to determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product

The required parameters to the solution are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, however, this number can be determined from the other parameters

### Assumptions

1. Demand for items from inventory is continuous and at a constant rate
2. Production runs to replenish inventory are made at regular intervals
3. During a production run, the production of items is continuous and at a constant rate
4. Production set-up/ordering cost is fixed (independent of quantity produced)
5. The lead time is fixed
6. The purchase price of the item is constant, i.e. no discount is available
7. The replenishment is made incrementally

### Variables

• K = ordering/setup cost per production run
• D = yearly demand rate
• h = yearly holding cost per product
• T = cycle length
• P = yearly production rate
• ${\displaystyle x={\frac {D}{P}}}$
• Q = order quantity

### Total cost function and derivation of EPQ formula

• Holding Cost per Year = ${\displaystyle {\frac {Q}{2}}h(1-x)}$

Where ${\displaystyle {\frac {Q}{2}}}$ is the average inventory level, and ${\displaystyle h(1-x)}$ is the average holding cost. Therefore, multiplying these two results in the holding cost per year.

• Ordering Cost per Year = ${\displaystyle {\frac {D}{Q}}K}$

Where ${\displaystyle {\frac {D}{Q}}}$ are the orders placed in a year, multiplied by K results in the ordering cost per year.

We can notice from the equations above that the total ordering cost decreases as the production quantity increases. Inversely, the total holding cost increases as the production quantity increases. Therefore, in order to get the optimal production quantity we need to set holding cost per year equal to ordering cost per year and solve for quantity (Q), which is the EPQ formula mentioned below. Ordering this quantity will result in the lowest total inventory cost per year.

EPQ formula

${\displaystyle Q={\sqrt {\frac {2KD}{h(1-x)}}}}$

### Relevant formulas

• Average holding cost per unit time:
${\displaystyle {\frac {1}{2}}hD(1-x)t}$
• Average ordering and holding cost as a function of time:
${\displaystyle x(t)={\frac {1}{2}}hD(1-x)t+{\frac {K}{t}}}$

The newsvendormodel is a mathematical model in operations management and applied economics used to determine optimal inventory levels. It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is , each unit of demand above is lost in potential sales. This model is also known as the newsvendor problem or newsboy problem by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

The economic lot scheduling problem (ELSP) is a problem in operations management and inventory theory that has been studied by a large number of researchers for more than 50 years. The term was first used in 1958 by professor Jack D. Rogers of Berkeley, who extended the economic order quantity model to the case where there are several products to be produced on the same machine, so that one must decide both the lot size for each product and when each lot should be produced. The method illustrated by Jack D. Rogers draws on a 1956 paper from Welch, W. Evert. The ELSP is a mathematical model of a common issue for almost any company or industry: planning what to manufacture, when to manufacture and how much to manufacture.

The reorder point (ROP) is the level of inventory which triggers an action to replenish that particular inventory stock. It is a minimum amount of an item which a firm holds in stock, such that, when stock falls to this amount, the item must be reordered. It is normally calculated as the forecast usage during the replenishment lead time plus safety stock. In the EOQ model, it was assumed that there is no time lag between ordering and procuring of materials. Therefore the reorder point for replenishing the stocks occurs at that level when the inventory level drops to zero and because instant delivery by suppliers, the stock level bounce back.

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