 Hello everybody, I am Dr. Keshav Valase from Valchin Institute of Technology, Solapur. In this session, we will discuss about further part of derivation towards production consumption inventory model. At the end of this session, students will be able to understand what are the different assumptions and variables in this particular production consumption inventory model as well as they will be able to understand what are different steps in deriving EOQ that is economic order quantity or EBQ that is economic batch quantity and associated inventory cost for this particular model that is production consumption inventory model. Now in earlier video, we have discussed these notations, but briefly we will revise it here. So here this capital Q represents holding quantity, point A projected on vertical axis is a graph between quantity and time. So capital Q is quantity physically holding, small Q is the quantity that we are manufacturing in a particular batch, R is consumption rate, K is production rate, K minus R indicates inventory built up rate that is this line from O to A. We manufacture at K, consume at R and the difference will indicate the rate at which we build the inventory till point A is raising to physically holding that is capital Q. Next is T1, this T1 is the time period wherein we produce as well as we consume the inventory items and T2 is the time period wherein we are just consuming we are not producing and O to B is the small t that is total cycle time. At this point, I expect the viewers to think of these combinations of physically holding quantity capital Q, small Q that is quantity manufactured what we call as the E B Q economic batch quantity, quantity manufactured actually and T the cycle time. So this capital Q, small Q the batch quantity and the cycle time. What could be the nature of these three parameters for a certain manufacturing industry? I expect you to just give a thought, think on this, you can take any industry say auto industry or machine tool manufacturing industry or any industry that you have in your mind you can think of manufacturing industry for a while. Now again, this we have discussed in previous video, I will just move fast on this. Here A B represents R as consumption rate, O A represents inventory built up rate, then small Q the batch quantity manufactured, the small Q is the batch quantity manufactured actually that is equal to the production rate into corresponding time of production and secondly, capital Q is the inventory built up rate K minus R into corresponding time T 1 and combining these two what we get here is capital Q is equal to K minus R upon K and multiplied here in the numerator again is small Q. Make the difference clear between these two Q's, this capital Q is the quantity of physically holding, small Q is the quantity of manufacture, I repeat small Q is the quantity we have manufactured and capital Q is after consumption how much physically it remains for holding. So, this we need for the derivation towards E B Q and inventory because we need this capital Q for calculating holding cost because this is the quantity holding, we cannot consider small Q to calculate holding cost in this particular model because capital Q is the holding quantity, small Q is not the holding quantity and hence from these two relations, we are coming up with this capital Q to small Q relationship which then will be using further derivation. Now, if we move in the steps for derivation, very first holding cost we will calculate holding cost is calculated by first referring unit holding cost which is specified as rupees per unit item per unit time. Hence this unit holding cost we multiply with average quantity now this quantity has to be the average holding, the quantity which we are holding and not the quantity manufactured. So, I repeat again this should be capital Q related calculations and not small Q related calculations because we are not holding small Q, we are holding capital Q in stores and again that we multiply by cycle time and then number of cycles in a year. This O S to B this is one cycle like this in a year we can have N cycles. So, here we multiply with that N. So, if we substitute the symbols C 1 is the unit holding cost capital Q by 2 represents the average holding quantity then cycle time is T and N is the number of cycles. So, this if we simplify we get holding cost is equal to 1 half into capital Q into C 1 because T into N is 1 we have discussed it earlier. Now, coming to some substitutions this relationship which we have discussed earlier. So, here this Q we need to replace with a small Q. Hence from equation 3 that we have got in earlier slide this capital Q we will replace with this relationship because we need this relation of cost in terms of a small Q that is a quantity manufactured. Hence holding cost will come out as equal to 1 half into K minus R into Q by K and outside we have multiplied by C 1. This is the replacement for capital Q. This is again put up in simplified version. So, this is ultimately equation 4 represents holding cost part of the derivation. Next is shortage cost with the assumption if we go back we have assume shortage is not permitted that is C 2 is 0 and hence C 2 will not appear in this derivation for production consumption inventory model as per the assumption. Next is setup cost. This setup cost C 3 is basically specified as cost per cycle. So, C 3 is the cost in rupees per cycle. So, that we need to multiply with number of cycles. Hence this setup cost comes out to be C 3 into N. Now, N is 1 upon T hence we again simplify this. So, setup cost will be C 3 upon T ultimately. Now here we add up these costs and finally put up the total inventory cost which is holding cost plus setup cost and the second C 2 is 0. Hence this C if we substitute for both these cost elements here from previous slide. This part represents the holding cost aspect and this part represents setup cost aspect. This is two parts are discussed in previous slide. Here one change we will again have for the substitution and that is this T we will replace by Q by R from the relationship the basic relationship of quantity and consumption rate we get this T as Q by R. So, that we are replacing here. So, now this equation number 6 is the cost equation which we need in terms of small Q that is batch quantity the quantity we manufacture. Now, this cost to be minimum we need to differentiate this cost equation with respect to small Q and equate it to 0. I repeat we need to differentiate this cost equation with respect to Q partially, partial differentiation we need here and equate it to 0 and then simplify that to get ultimately what is this small Q. Once we simplify that after the derivation we get this Q as square root of 2 C 3 R upon C 1 and again a bracket this all goes under the square root K minus R upon K. So, this equation 7 represents Q star we put a star here because this now becomes economic batch quantity and this is C minimum which we get by putting this Q back to the cost equation and that is how we get this minimum cost associated with economic batch quantity. Because equation 7 and 8 are the two parameters which we are ultimately interested in this total cost curve the lowest point which gives us the OQ or ABQ in case of manufacturing and associated minimum cost or production consumption inventory model. This again I recommend these two books of S. D. Sharma and Hira Gupta for the references. Thank you.