Question #186878

The demand for a certain product is 2000 units per year and the items are withdrawn at a constant rate.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. 


1
Expert's answer
2021-05-07T11:47:05-0400

Let q= numbers of unit ordered at a time

  

Total cost = commodity cost + ordering cost + holding cost


  T=C.C.+O.C.+h.c.T=C.C.+O.C.+h.c.

     

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

            =2000×470=940000= 2000\times 470=940000

 

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


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


           =4.1×Q+D2= 4.1\times \dfrac{Q+D}{2}


 (a) TC=2000×+50×2000Q+4.1Q2TC= 2000\times +\dfrac{50\times 2000}{Q}+\dfrac{4.1Q}{2}


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


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


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


          0100000q2+2.05=0\Rightarrow 0-\dfrac{100000}{q^2}+2.05=0


           q=1000002.05=220 units q=\sqrt{\dfrac{100000}{2.05}}=220 \text{ units }


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


           =QUyear=12×20.862000=1.325 months =\dfrac{Q}{U} year=\dfrac{12\times 20.86}{2000}=1.325 \text{ months }


  Hence order shouls be placed after 1.325 months.


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