EOQ with price gap

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Russian: Экономичный размер запаса с разрывами цен

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;
Lead time is zero.

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: $\, 0 \le y <y_ {\ min}$ , $\, y_ {\ min} \le y <q ^ {* }$ , $\, y> q ^ {*}$ , 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: