# Economic production lot size

Economic production lot size model represents the quantity of items of goods produced or purchased at the lowest total inventory costs. It can be achieved by balancing decreasing unit’s setup cost and increasing unit’s inventory cost. This model was developed in 1918 by E.W. Taft. This method continues the economic order quantity model (EOQ model).  It differs from EOQ model because Economic production lot size model assumes that the company (typically, manufacturer) produces product by itself, therefore the goods are received incrementally while they are produced. On the other hand, EOQ model assumes that the ordered quantity arrives complete after ordering, because the products are produced by another company and are ready to delivery when the order is received.

## Overview

Economic production lot size model is applicable only when the demand for a product is constant throughout the year and when each new order is produced incrementally when the inventory reaches zero. The model divides cost into three parts: setup costs, inventory costs and direct costs (directly proportional the quantity of goods produced, include materials and labor costs).

## Economic production lot size model

To develop the model several assumptions should be made:

• Constant demand exists throughout the year;
• The purchase price is fixed (no discounts available);
• The lead time is fixed;
• Productions runs to replenish inventory are made at regular intervals;
• Incremental replenishment;
• Production setup cost is fixed (does not depend on quantity produced);
• Continuous production exists during the production run. Production is also kept at the constant rate.

In order to determine the optimal number of units to produce we have to minimize the total inventory cost. The required parameters for the model are following : K – ordering/setup cost; D – demand rate; F – holding (inventory) cost; T – cycle length; P – production rate; x=D\P; Q – order quantity. Thus, holding cost per year may be calculated according to the formula: Holding costs = Q/2*F(1-x); Where is the average inventory level throughout the year, and F(1-x) is the average holding cost. Setup cost can be calculated in the following way: Ordering cost per year = D/Q*K; Where is the number of orders placed in a year, multiplied by K equals in the ordering cost per year. From the equations above follows that total ordering (setup) cost decreases with the increase in production quantity. However, the total inventory cost increases with the increase of production quantity. Therefore, in order to get the economic production lot size we have to equate inventory cost to setup cost and solve for quantity (Q), which gives us economic production lot size formula below. Ordering this quantity results in the lowest total inventory cost per year.

The graph gives better understanding of the economic production lot size . The upper line results from addition of the rest and represents total inventory cost. When stepping aside from the optimal EPL level, several problems may be faced.

When ordering higher quantity:

• Low setup costs and high inventory costs;
• Higher production efficiencies are fostered;
• Capacity increases and utilization is reduced.

When ordering low quantities:

• High setup costs and low inventory costs;
• Negative impact on production efficiencies;
• Capacity decreases and utilization increases.

The mathematical model presented above appears to be quite simple but the correct inputs should be taken in order to get correct results. As for the setup costs, using all the costs of receiving and purchasing appears to give an exaggerated answer. (Piasecki, 2003)

## Setup cost

This is a sum of fixed cost for each purchased item. These costs are associated not with quantity that is ordered but rather with activities that are involved in processing the order. For the case of manufacturing that matters in this certain case, this type of cost will include the time for work order initiation, the time for picking and issuing some components. However, the time for counting and handling specific quantities is excluded as well as the time for scheduling the product, setting up the machines and inspecting the project.

## Inventory costs

This type of costs is associated with having the inventory in stores. It consists of investment in inventory and with costs of storage. The primary components of the Inventory costs are Interest (If one has to borrow money to hold the inventory, or one has loans. If one is debt free the amount of money that could be made by investments should be counted), Insurance (It is directly related to the total inventory value), Taxes and storage Costs (Here only costs that are variable based on the level of inventory are included). 3.