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:

Z2pic1.JPG

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:


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

E2.1.jpg

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


E2.2.jpg

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


E2.3.jpg

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 }.

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