 Hello everybody. I am Dr. Keshav Valase from Walton Institute of Technology, in this session we will discuss about production consumption inventory model but particularly through numerical aspects. At the end of this session students are expected to understand what is the concept of economic order quantity or economic batch quantity as well as associated inventory costs for this particular model we have selected that is production and consumption type of inventory model. At the same time in numerical perspective students are expected to understand the concept of what it really means with production and consumption inventory model. Now these things are discussed at length in earlier videos of derivation. Here it is just again we are revisiting the symbols and notations. This we are not really discussing here because stepwise details are given in earlier video in concise form only I will just mention here that this is the cost equation wherein this first term from here 1 half q c 1 and till up to this k this corresponds to the holding cost term and this second part c 3 by d it corresponds to the setup cost. Here c 2 we are not considering because in the assumptions holding cost is assumed to be 0. This is the cost equation again in altered form t replaced in terms of small q that is batch size. Here in earlier holding cost calculations also we had capital q that was holding quantity which is again replaced in terms of small q and for setup cost t is again replaced in terms of small q because we are basically interested in knowing what is small q that is what quantity we should manufacture and deriving this cost equation partially with respect to the parameter q small q and equating that to 0 for the cost to be minimum and solving that equation we ultimately get this relation of small q star which we call as economic batch quantity which is given with this particular relation in equation number 2. Next is minimum cost associated with this e b q substituting this q star back to this cost equation number 1 and simplifying we get that cost which is minimum cost hence named as c minimum this cost with formula number 3. So, all these things we have revised again here. Now we aim at understanding the numerical part that is understanding the model with some numbers so that sometimes without numericals understanding the concept with only symbols becomes little difficult. So, I intended to refer some numbers and explain or discuss what is the real meaning of a production and consumption and type of inventory model. Now here let us assume this k capital K which is production rate let us assume this production rate as 60 units per day that is we are producing 60 units in one day. Secondly consumption rate indicated by symbol r let us assume it as 20 units per day that is we are producing 60 units in a day and we are consuming 20 units in a day. Remember in earlier video we have discussed this k must be greater than r rate of production has to be more than rate of consumption then only we will be able to reach to some capital Q the holding quantity to point a. Hence this k we are assuming 60 per day whereas r the consumption rate we are assuming as 20 units per day we are consuming. Now with this for the simplicity of discussion and understanding we take this T 1 in the graph if you take this T 1 is the time of production as well as consumption from origin to C point this is a T 1 is equal to 10. Now this 10 days I repeat again we are manufacturing the quantity. So, this line indicates inventory built up at the rate of k minus r then T 2 is 20 days it is this particular time from C to B the second part of the cycle point C to point B it is a T 2 which is a 20 during this cycle during this period that is from C to B a long line A to B we are only consuming the quantity which was raised to point A that is Q capital Q. So, this quantity we will be now consuming down to 0 till point B. So, now with these assumptions we will again go ahead and try to understand what it is. Before that I expect you all people to think about the different values of k and r that is different values of production rate k and consumption rate r takes some different cases different values assume these also different values and that in case of any manufacturing industry that you think of may be in auto industry or any fan manufacturing machine tool manufacturing industry. For such industry in practical application side you just think of a case and how does it reflect for that particular industry you are thinking. Now going ahead these are the things we have assumed we have discussed earlier and now capital Q that is this quantity capital Q we get with the rate at which we physically build the inventory remember I repeat again we are manufacturing 60 units per day and we are consuming 20 units per day the difference of these two this k minus r is nothing but the inventory built up physically along this line at the rate of k minus r here it is equal to 40 at the rate of 40 per day we are building the inventory physically for holding in the store and that we are doing it for T 1 time period. So, mathematically in symbolic notations capital Q comes out to be k minus r in the bracket into T 1 which we substitute the values it gives 14 to 10 is equal to 400 units. So, this is the quantity we are holding physically we are holding 400 units if we see here point A corresponds to quantity capital Q which is 400 physically holding but actually how much we have manufactured that is a small Q small Q we can get with capital K that is production rate into time of production time of production is this T 1 which is equal to 10 thus this small Q is equal to 60 into 10 that gives us 600 units the quantity manufactured I make the difference clear here again this small Q is the quantity actually we have manufactured and this 400 capital Q is the quantity physically we are holding going next little further ahead these two things we have discussed capital Q as 400 small Q as 600 we are manufactured 600 but we are physically holding 400. Now, let us look at what is happening to consumption during T 1 now consumption during T 1 if you want to calculate that is during this T 1 period in 10 days how much we are consuming our consumption rate is 20 per day and we are now considering T 1 time period. So, you multiply R and T 1 that will give us the consumption during time T 1 which is 20 into 10 is equal to 200 similarly consumption during T 2 time period that is during 20 days from C to B it will be consumption rate R into corresponding time T 2. So, here both are 20 into 20 that gives us 400. So, this 400 will be consuming it down to 0 thus in a nutshell if we talk small Q is the quantity we actually manufactured 600 units we actually manufactured out of that 200 units we consumed while we were producing during T 1 600 minus 200 it will be 400. So, this 400 is the actual physical holding quantity here and at this point A we are stopping the manufacturing and then we only consume along the line A to B. So, I repeat again 0.0 to A we are producing as well as consuming whereas, from point A to B we are only consuming the quantity which was raised to 400 thus the name comes up production consumption inventory model. So, ultimately in T 1 time period we consume 200 units in T 2 time period what are the 400 holding was there that 400 we consumed down to B that is to 0 and then again the same cycle repeats. Thus what are the values we calculated we can refer this E b Q that is Q star and C minimum formula using C 1, C 2, C 3 here C 2 is 0, but otherwise in general we can substitute the values here and calculate these two main parameters which are indicated in this total cost versus quantity graph. So, this E o Q or E b Q corresponds to this formula, this value that is this is economic is Q star and this corresponds to minimum cost. Point to be taken care of is R and C 1 both must be in same time units. These are the two books I recommend here. Thank you.