# Basic economic order quantity

Russian: Экономичный размер заказа

Let us consider the case of static demand.

## The basic assumptions of the model

• Number of units consumed per period is a constant (demand rate);
• Price of the resource is constant;
• Carrying cost of the resource is constant;
• Order cost is constant;

## The basic notations

• $\, D$ – demand rate;
• $\, h$ – marginal carrying costs;
• $\, K$ – fixed order costs;
• $\, t_{0}$ – order cycle time;
• $\, TCU(y)$ – total costs per period;
• $\, y$ – order quantity;
• $\, y^{*}$ – economic order quantity.

Inventory optimal control

According to the assumptions dynamics of inventory takes the following form (Fig. 1):

Fig. 1. The dynamics of inventory in the economic order quantity model.

Since the intensity of demand for the resource is constant, the average inventory level is $\, \frac {y} {2}$ units. Therefore, the total cost per unit time can be explained as a function of order volume in the form of the cost of ordering a unit of time and storage costs of the resource per unit time:

(1) $TCU(y)=\frac{K}{t_{0} } +h\left(\frac{y}{2} \right)t_{0}$ .

Given the dependence of the order cycle time, the intensity of demand for the resource, $\, t_ {0} = \frac {y} {D}$, equation (1) becomes:

(2) $TCU(y)=\frac{KD}{y} +h\left(\frac{y}{2} \right)$ . The first order condition for the function (2) becomes:

(3) $\frac{d}{dy} \left(TCU(y)\right)=-\frac{KD}{y^{2} } +\frac{h}{2} =0$ . The second order condition for the function (2) is:

(4) $\frac{d^{2} }{dy^{2} } \left(TCU(y)\right)=\frac{2KD}{y^{3} }$ .

Therefore, the function $\, TCU(y)$ is convex with respect to $\, y$, where $\, y>0$ . Then, the solution of equation (3) of the form

$y^{*} =\sqrt{\frac{2KD}{h} } >0$

is the minimum point of $\, TCU (y)$. Value $\, y ^{*}$ is the economic order quantity.

The optimal duration of order cycle becomes:

(5) $\, t_{0}^{*} =\frac{y^{*}}{D}=\frac{1}{D}\sqrt{\frac{2KD}{h}}=\sqrt{\frac{2K}{Dh}}$

Total minimum costs per period, $\, TCU^{*}$ , takes the following form:

(6) $TCU^{*} =TCU(y^{*} )=\frac{KD}{y^{*} } +h\left(\frac{y^{*} }{2} \right)=$

$=KD\sqrt{\frac{h}{2KD} } +\frac{h}{2} \sqrt{\frac{2KD}{h} } =\sqrt{\frac{KDh}{2} } +\sqrt{\frac{KDh}{2} } =$

$=\, \sqrt{2KDh}$ .

Optimal inventory control in the economic order quantity is the following:

How much to order?. Order $y^{*} =\sqrt{\frac{2KD}{h} }$ units of the resource.

When to order? After each $t_{0}^{*} =\frac{y^{*} }{D}$ units of time.

## Example.

On an assembly line of computers daily are consumed 50 processors. The cost of placing an order for the purchase of processors regardless of how much the party is $25. Carrying cost of a single processor per day is$ 0.25. What is the optimal inventory control strategy.

Solution.

$\, D=50$ processors per day,

$\, h= 0,25$ for carrying one processor,

$\, K= 25$ for order.

Then economic order quantity is

$y^{*} =\sqrt{\frac{2KD}{h} } =\sqrt{\frac{2\cdot 25\cdot 50}{0,25} } =100$

processors, and the optimal order cycle time is

$t_{0}^{*} =\frac{y^{*} }{D} =\frac{100}{50} =2$ days.