Answer to Question #186878 in Operations Research for hami

Question

The demand for a certain product is 2000 units per year and the items are withdrawn at a constant rate. The ordering cost incurred each time an order is placed to replenish inventory is £50. The unit cost of purchasing the product is £470 per item, and the holding cost is £4.10 per item per year.

Apply a basic inventory model to determine the optimal size of each order and how often an order should be placed. You should follow the following steps:

(a) Formulate the mathematical problem.

(b) Determine the optimal size of each order.

(c) Determine how often an order should be placed. 

Answer 1

Let q= numbers of unit ordered at a time

  

Total cost = commodity cost + ordering cost + holding cost

  $T=C.C.+O.C.+h.c.$

     

  Commodity cost (CC)= units per year \times cost per unit 

            $= 2000\times 470=940000$

 

  Ordering cost $(OC)=\dfrac{50\times 2000}{Q}$

  Holding cost (HC) = cost of carrying$\times$ average inventory

           $= 4.1\times \dfrac{Q+D}{2}$

 (a) $TC= 2000\times +\dfrac{50\times 2000}{Q}+\dfrac{4.1Q}{2}$

     $= 94000+\dfrac{100000}{Q}+2.05Q$

(b) For optimum size $\dfrac{d(TC)}{dq}=0$

           $\dfrac{d}{dq}(94000+\dfrac{10000}{q}+2.05q)=0$

          $\Rightarrow 0-\dfrac{100000}{q^2}+2.05=0$

           $q=\sqrt{\dfrac{100000}{2.05}}=220 \text{ units }$

(c) Length of production run/ or length of order $=\dfrac{1}{\text{ NO. of order}}=\dfrac{1}{\frac{U}{Q}}$

           $=\dfrac{Q}{U} year=\dfrac{12\times 20.86}{2000}=1.325 \text{ months }$

  Hence order shouls be placed after 1.325 months.