# EOQ with price gap

Consider the case of static demand.

## The Basic Assumptions of the Model

• Number of units consumed per unit of time is a constant (demand rate);
• Price of the resource unit depends on order volume. If the order quantity does not exceed a certain level of $\, q ^{*}$, then the price is constant $\, c_ {1}$, otherwise - the price is a constant $\, c_ {2}$, where $\, c_ {1}> c_ {2}$:

$c=\left\{\begin{array}{l} {c_{1} ,\, \, y\le q^{*} ,} \\ {c_{2} ,\, \, y>q^{*} ;} \end{array}\right.$

• Carrying cost of resource unit is a constant;
• Order cost is a constant;

## The Basic Notation

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

## Inventory Optimal Control

Costs of resource per unit of time are a function of the size of orders:

$\left\{\begin{array}{l} {c_{1} \frac{y}{t_{0} } ,\, \, y\le q^{*} ,} \\ {c_{2} \frac{y}{t_{0} } ,\, \, y>q^{*} .} \end{array}\right.$

Given the dependence of the order cycle time on the intensity of demand for the resource, $\, t_{0} = \frac {y} {D}$, the cost of the purchase of products per unit time can be represented as:

$\left\{\begin{array}{l} {c_{1} \frac{y}{\left(\frac{y}{D} \right)} =Dc_{1} ,\, \, y\le q^{*} ,} \\ {c_{2} \frac{y}{\left(\frac{y}{D} \right)} =Dc_{2} ,\, \, y>q^{*} .} \end{array}\right.$

The total cost per unit time can be represented as a function of order volume $\, y$ as the sum of acquisition costs of resource per unit of time, the cost of ordering and storage costs of the resource per unit time:

$\,TCU(y)=\left\{\begin{array}{l} {TCU_{1} (y)=Dc_{1} +\frac{K}{t_{0} } +h\left(\frac{y}{2} \right)t_{0} ,\, \, y\le q^{*} ,} \\ {TCU_{2} (y)=Dc_{2} +\frac{K}{t_{0} } +h\left(\frac{y}{2} \right)t_{0} ,\, \, y>q^{*} ,} \end{array}\right.$

or:

$TCU(y)=\left\{\begin{array}{l} {TCU_{1} (y)=Dc_{1} +\frac{KD}{y} +h\left(\frac{y}{2} \right)t_{0} ,\, \, y\le q^{*} ,} \\ {TCU_{2} (y)=Dc_{2} +\frac{KD}{y} +h\left(\frac{y}{2} \right)t_{0} ,\, \, y>q^{*} ,} \end{array}\right.$

The graphs of the function $\, TCU_{1} (y)$ and $\, TCU_{2} (y)$ are the following, fig. 1:

Fig. 1. The function of total costs

The point $\, y_{\min }$ is determined according to economic order quantity:

$y_{\min } =\sqrt{\frac{2KD}{h} } >0$.

In the point $\, q^{*}$ the following inequality holds:

$\, TCU_{1} (y_{\min } )=TCU_{2} (q^{*} )$,

or:

$TCU_{1} (y_{\min } )=Dc_{2} +\frac{KD}{q^{*} } +h\left(\frac{q^{*} }{2} \right)t_{0}$,

then:

$(q^{*} )^{2} +\left(\frac{2(Dc_{2} -TCU(y_{\min } ))}{h} \right)q^{*} +\frac{2KD}{h} =0$.

When $\, y \le q ^{*}$ graph $\, TCU (y)$ equal to $\, TCU_ {1} (y)$. When $\, y> q ^{*}$ graph $\, TCU (y)$ equal to $\, TCU_ {2} (y)$. In correspondence with a graph, consider the three areas on the x-axis: $\, 0 \le y , $\, y_ {\ min} \le y , $\, y> q ^ {*}$, which are called, respectively: A, B and C. The optimum size of the order $\, y ^ {*}$ depends on what area is the point $\, q ^ {*}$, Fig. 2, 3, 4:

$y^{*} =\left\{\begin{array}{ll} {y_{\min } ,} & {q\in A} \\ {q,} & {q\in B} \\ {y_{\min } ,} & {q\in C} \end{array}\right.$

Fig. 2. $\, q\in A$ , $\, y^{*} =y_{\min }$.

Fig. 3. $\, q\in B$ , $\, y^{*} =q$ .

Fig. 4. $\, q\in C$ , $\, y^{*} =y_{\min }$.

Optimal inventory control in the model is the following:

Step 1. According to EOQ compute $y_{\min } =\sqrt{\frac{2KD}{h} } >0$ . If $\, q\in A$ , then $\, y^{*} =y_{\min }$ , else, go to step 2.

Step 2. Compute $\, q^{*}$ from the following equation:

$(q^{*} )^{2} +\left(\frac{2(Dc_{2} -TCU(y_{\min } ))}{h} \right)q^{*} +\frac{2KD}{h} =0$ ,

to determine the border areas $\, B$ and $\, C$ . If $\, q\in B$ , then $\, y^{*} =q$ , if $\, q\in C$ , then $\, y^{*} =y_{\min }$.