 Hello everybody, I am Mr. Valasek Keshav from Mechanical Engineering Department of Valchan Institute of Technology, Sualapur. In this video, we will be discussing about EOQ that is Economic Order Quantity, its derivation for a very simple model. At the end of this session, students are expected to understand what is the concept of Economic Order Quantity as well as associated minimum inventory cost and the steps in the derivation of EOQ as well as inventory cost. This particular slide we have discussed in earlier video part 1. Here this graph indicates on vertical axis inventory cost and on horizontal axis it indicates inventory quantity. This straight line represents holding cost or inventory carrying cost whereas this curved line indicates setup cost or ordering cost. Summation of these two is represented by this U shape line that is total cost. This has been discussed in earlier video. Now here I would like to think you all about the different types of costs we have discussed in previous video. Those we need here for the derivation, they are holding cost, setup cost and shortage cost. I expect you all at this stage to think for a while and compare these different types of inventory costs. Coming to the derivation steps, this is a very simple model for EOQ derivation wherein the assumptions are mentioned here and these are some main terms we need to refer in the derivation. This Q is the quantity whereas this T on horizontal axis indicates the time scale. So this in all triangle indicates it represents one particular inventory cycle and this land line is R that is consumption rate. Assumptions as mentioned here just to revise again as against earlier video. This R that is consumption rate is assumed to be uniform that is if this quantity if we take a suppose 300 for a cycle and if this time period if we take as 30 days for one month suppose then this R consumption rate will be Q that is 300 divided by T that is 30 giving us R is equal to 10 units per day. This is what we assume this R to be uniform. Secondly, no shortages permitted that is symbol if we consider C2 we denote as shortage cost and that will be 0. Late time is 0 or known exactly this is discussed earlier as well as instantaneous replenishment that is at the end of this cycle we assume that for the next second cycle we have quantity Q available next day we come for the business. So that is the assumption that any part of time we need the quantity to be raised to its level small Q it is available. Now, coming to the steps in the derivation here we start with holding cost you know all the cost inventory related cost is the sum of all the cost here and to begin with we let us start with the holding cost. So, this holding cost can be taken up as unit holding cost multiplied by average quantity multiplied by its cycle time into number of cycles in a year or symbolically unit holding cost we represent with symbol C1 which is rupees per unit item per unit time. So, this C1 is represented unit cost is represented for one item and for unit time. So, we multiply this by the quantity of a cycle which here it comes out to be one half Q because this quantity from certain maximum Q level drops down to 0 level. Hence, we assume that average quantity one half Q is available all over the period of cycle that is cycle time t. Hence, here we are considering quantity as one half Q that is average quantity. Now, this multiplied by this cycle time t which is here and like this we have in all n cycles in one year. So, this we again multiply by n that is number of cycles in a year. If we simplify this we get this holding cost is equal to one half into small Q into C1. Now, here t into n becomes 1 because cycle time t is a reciprocal of number of cycles here. Now, secondly the shortage cost as we assumed it to be 0 no shortage is permitted in the assumption we have mentioned that. So, shortage cost will be 0. Third element set up cost basic set up cost C3 is mentioned as a rupees per cycle. So, to get the set up cost we need to multiply unit set up cost by number of cycles. So, we get it here set up cost is equal to C3 which is per cycle into n number of cycles. Again as said here this n will be reciprocal of t 1 upon t. Now, these two equations this equation for holding cost and secondly equation number 2 for set up cost these two equations we will need for further derivation. So, from one and two from previous slide total inventory cost will be holding cost plus set up cost here shortage cost is 0. So, this C if we take the symbol for total inventory cost is equal to one half q into C1 which represents holding cost plus C3 upon t which represents set up cost. Now, here again this t we can replace by q upon r and finally we get this as the total cost equation which is the sum of holding cost as well as set up cost. I repeat shortage cost C2 does not come here because we have assumed that to be 0. Now, this cost equation we need to partially differentiate with respect to small q for this cost to be minimum. Hence, if we differentiate this C with respect to small q we get q is equal to square root of 2 into C3 into r upon C1. So, this q star we call as EOQ after differentiation of this cost equation whatever the quantity we get that we have represented here as q star meaning that it is economic order quantity and the formula for that we have got is square root of twice C3 into r divided by C1. Similarly, this q star if we put back in this equation then we will get inventory cost which will be minimum corresponding to this economic order quantity. Thus substituting this q star in this cost equation number 3 we get and then simplifying of course we get minimum inventory cost is equal to square root of twice into C1 into C3 into r. So, this is how in all we have got two terms if we again compile that in the graph on total cost curve as discussed earlier in previous video this low most point is the point of our interest wherein on horizontal axis it represents economic order quantity and this is given by this formula EOQ is equal to q star which is equal to square root of twice C3 into r divided by C1 and for this low most point on the total cost curve on vertical axis this point represents the minimum inventory cost that is symbolically we indicate it as C minimum. So, this C minimum mathematically we get here as C minimum is equal to square root of twice C1 into C3 into r. So, this is how we have derived the equations for economic order quantity and minimum cost for a very simple model in inventory. Here as a reference I just mentioned two books but we have ample books in operations research in general and even for inventory control. Thank you.