 Hello everybody, I am Dr. Keshav Wallase from Valchand Institute of Technology, Solapur. In this session, we will be discussing about production consumption inventory model, its preliminary things required for the derivation. The terms of this session will be like viewers will be in a position to understand what are the different variables and assumptions in defining production consumption type of inventory model. Again, they will be able to understand overall idea of this particular model and some preliminary calculations, the steps towards the derivation of economic batch quantity and associated inventory costs in case of this model of production consumption inventory model. Let us revise briefly about the types of models in inventory control. Very broadly, there are three types or three groups of inventory models, deterministic models, probabilistic models and models with discounts. We have discussed these things in detail in earlier videos and here we will be talking about deterministic models only and in that we have discussed again that any model making depends upon the assumptions of the variables, particularly here, shortage cost which is denoted as C2 and secondly the production rate which is symbolically given as K. The way we assume these two variables will define our inventory model, we are confining our discussions towards deterministic models as I said, hence whether the shortages are permitted or not permitted that gives us the models with or without shortage cost and second is about K whether this K that is production rate is assumed to be finite or infinite. Because the combinations of these two assumptions for these two variables in all leads to four deterministic types of inventory control models. Here I would expect the students to think about the application part of these models in the sense the suitability of inventory models those come up with the combinations of the assumptions. Coming to this particular type of model will be talking in detail about production consumption inventory model, firstly we will talk about the notations, graphically this model is represented here where in vertical line indicates the quantity, horizontal line the horizontal axis indicates time, so this is the graph between quantity versus time to represent any inventory model. Now, here if you talk of notations the symbols capital Q is the holding quantity that is the quantity that we are physically holding on the graph that is depicted with point A, the point A on projected on vertical axis would give us here this physically holding quantity capital Q. Secondly small Q is the quantity manufactured in a particular batch, this R is the consumption rate at what rate we are consuming the inventory, K is the rate of production and now here K minus R is indicated as inventory built up rate which is this line from O to A here in K is the rate of production which is higher than rate of consumption R and whatever we are producing K from that we are consuming certain quantity at the rate of R and remaining K minus R quantity would indicate the rate at which we built up the inventory till point A which ultimately gives us capital Q as the holding quantity physically holding quantity. Finally T 1 is the time that is from O to this C point this time period is the time period wherein we are producing we are a manufacturing as well as we are consuming the inventory items. Here I repeat during time period T 1 we are producing as well as consuming the quantity while we reach point A we stop manufacturing and thereafter whatever the quantity is physically holding capital Q that we are consuming we are only consuming then from A to B at the rate of R for the time period T 2. So, T 2 is the time period of only consumption during this we are not manufacturing anything thus in all O 2 B is the total cycle time. So, graphically this is in all how the model of production consumption inventory control model looks like. Now, let us briefly discuss about the assumptions as discussed earlier any inventory control model comes up with the set of assumptions and this particular model two basic assumptions are highlighted here first is shortages are not permitted no shortages that is mathematically C 2 is 0. Secondly production rate K is assumed to be finite or in other words it is also said as gradual replenishment of inventory items. So, because of this assumption we are getting this particular line O 2 A otherwise we would have got this straight line. So, here as we are assuming K as a finite we are getting this line O 2 A next is consumption rate R that is assumed to be uniform. Next lead time L is either 0 or it is known exactly that is the assumption again here similarly buffer stock or the safety stock the stock which is referred to meet the uncertainty in the demand. So, that buffer stock is again assumed to be either 0 or known exactly. Now, we will proceed towards some basic calculations required before we finally go to the derivation of economic batch quantity for this model. Line AB here again if we talk represents R that is consumption rate line O A from origin to this highest point that represents inventory built up rate. We have discussed earlier this is the difference between at what rate we produce the inventory that is K minus the rate at which we consume the inventory that is R. So, K minus R the net it will represent inventory built up rate the rate at which we physically raise the quantity to capital Q as holding maximum quantity. Now, with the basic relationship of quantity and the rates and the time small Q is the quantity produced which is equal to K the production rate into the corresponding time of production. Now, here small Q I repeat this is the quantity produced which is equal to production rate into production time. So, T 1 here is a time period during which we are manufacturing hence K if we multiply with T 1 we get small Q that is the quantity actually produced. Secondly from triangle O A C we can write capital Q that is this vertical line which represents capital Q as a physical quantity capital Q is equal to its inventory built up rate K minus R into corresponding time of manufacturing T 1. So, capital Q here in equation 1 this small Q is the quantity manufactured make a difference clear I repeat again from first equation this small Q is the quantity manufactured in second equation capital Q is the quantity physically holding this is equal to the inventory built up rate into corresponding time T 1. Now, if you see here this was T 1. So, capital Q is equal to K minus R into small Q upon K this is we are replacing T 1. So, this relation of capital Q and small Q while we put in derivation steps for economy batch quantity we will be able to get the relationships we need. If we conclude here then this is a graph between inventory cost versus inventory quantity. Now, this total cost curve if we see here this is the point of our interest here in this is E O Q or E B Q in case of manufacturing and associated minimum inventory cost. So, this is the point which we need we are interested in and that is what we will be deriving economy batch quantity or economic order quantity and associated minimum inventory cost that we will discuss in next video. These are the two books I recommend one from S.G. Sharma and second by Hiran Gupta. Thank you.