 Hello everyone, welcome to the session of business simulation and risk analytics. In the previous session of this module, we have discussed the basic of Monte Carlo simulation and the importance of business simulation in business forecasting and also couple of illustrative example of Monte Carlo simulation. Today we will extend that concept to the continuous version of Monte Carlo simulation. That means in the previous session, we discussed some discrete event simulations or say frequency distribution are being captured in a discrete manner and then we have created the histogram and we have developed the simulation scheme, the Monte Carlo simulation scheme and then we have generated random number and we got different illustration through the simulation experiment. So, that was for discrete data set, today we will concentrate on continuous version of examples. So, if you recap that basic definitions within a first 2-3 minutes, so what we have done is that what is business simulation, what simulation helps you in understanding system behavior or the dynamic nature of the problem to get a deep system thinking, complex system thinking of the problem and you get a insight of it and you interpret the operational activities or the movement of the problem through different imitating process or repetitive process. So, that mimicking process or nothing but the simulation, same business problem or same behavior you are studying over a period of time or over different instances you can say. So, there are different type of simulation process are there and through that you understand the different insights or the business behavior of the system. So, it also helps you not only the study of system behavior, it also helps you the risk modeling part. Today we will discuss little bit about risk modeling part through Monte Carlo simulation of different business problem. At least you know one or two problem today we will study, then in the next session we will discuss more detail about the software illustration using advanced excel and then how we can make risk modeling using simulation of different business problem. And then recall the basic Monte Carlo simulation what we have discussed just previous session. So, it is nothing but generation of different random number through probability distribution of the particular variable. For example, if you have a variable say you know in a system this is a system say and in that case suppose you have a variable say or parameter say which is uncertain or risk and that you are generating and you are getting output of the system for a different input data that input data you directly do not give to the system you generate through random sample random number. This is what the Monte Carlo simulation and that process you repeat again and again and you generate different instances. This is what the Monte Carlo simulation in detail we have discussed and that can be generated through a whole process of Monte Carlo simulation or through a random function which we have discussed in the previous session. I will show you today also how to generate random function in excel again. These are the steps remember set the probability distribution of the data of the parameter that you are going to discuss or you are going to give as input of to your simulation model and then construct the cumulative distribution function second step and then generate the random interval for discrete case we have discussed in detail just recapping the process and then generate the random number from that particular interval and conduct the experiment unless you reach your threshold level or the stopping criteria and record all the instances and do the statistical inferences or sensitivity analysis to get insights of their problem and the system behavior and you may get many solutions of your problem. Now let us go to the continuous version of continuous case of simulation where the input data will not be in a discrete. So far 2-3 examples I have given including the previous sessions where all these data sets are considered as a discrete event right event and the corresponding frequency and the probability and you are generating cumulative and the interval which is very easy to understand and we have illustrated it effectively. But in case interval time of customers innovating lines say follow exponential distribution and you want to illustrate that through simulations. So in that case your distribution function should be exponential type distribution function right and then your density function will be you know exponential type and then you have to take the cumulative distribution functions you have to take. So in that case what will happen you have to calculate your capital effects and then the interval how will calculate the interval here because it is a continuous case right continuous case. And in that case you cannot create your interval because random number interval are not given here. So how to create the scheme for that that we are going to discuss this can be applied to exponential this can be applied to triangular because in practice the parameters value can be defined as uncertain or risk parameter through triangular distribution normal distribution which is very common in practice uniform distribution all these things can be done also. I will show you couple of applications today. So the parameter or input data sets can be of any type of distribution in case it is continuous in practice it is more practical right. So how to illustrate and create the Monte Carlo scheme that we are going to discuss how to create the Monte Carlo scheme this we are going to discuss for continuous case now. So let us go to this 1x illustration of say exponential distribution then rest we can understand through excel first understand how to create the Monte Carlo scheme. So let me write down the question now how to create the Monte Carlo scheme. So that we are going to discuss now. So first step the distribution function which you have already calculated you have assumed based on the data you pattern in the software I will show you how to generate or how to assign a distribution function to a data sets or what type of distribution that particular data set follow that also you can generate immediately through this software that risk address software. I will show you if not today in the next session but suppose interval of customers in a waiting line analysis follow exponential distribution you all know. So we are assuming that say fx which is of exponential type and the function we have assumed like this say right and the cumulative will be like that how come you can get the calculative you can calculate the cumulative the integrations 0 to x fxd. We know the range of this fx will be it is a cumulative value so it will be say you know general 0 basically x less because probability cannot be more than 1 so we will define it like this. So we know the range. And calculate it will be like you know 1 minus e to the power of minus lambda x you calculate this you can check at home and you can get to know how to calculate this. So once you will get this fx value we will enter into the step 3 say that means the interval but how will generate the interval because here it is a continuous case for the sake of illustration we are assuming it as uc. So in that case what we are going to do let us first you know you rearrange the data and we will get the exponential distribution and we will create the Monte Carlo scheme say random number interval scheme. So in that case what you will do I will take say rearrange say e to the power of minus lambda x equals to 1 minus u I have rearranged it say and then I will take ln of minus lambda x equals to ln of 1 minus u. I can take that both side I have taken. So it will be minus lambda x equals to ln of 1 minus u. Now x equals to minus 1 by lambda ln of 1 minus u but what is the range of u since 0 less than u less than 1 say or you know you can consider 1 of 2. So in that case since the random interval in case you consider 0 in that case it will be like 0 0 to 99 say we consider or say 1 to say 90 100 the way in discrete case I have talked. So you can consider accordingly whether you will consider the initial value or the last value. So accordingly you can put the less than greater than sign suppose because the total 100 number you have to allocate in a bucket say. So make sure that you should not count 101 you count only 100 number for the sake of 2 digits. If you go to n digits 3 digits it will be total 1000 number. So this way you can you know in that case if you take 0 0 0 then it should be 999. So that you can get 1000 number or 0 0 to 99. So this way you should get because all number are equally likely and whatever the number you pick from the bucket or random number you generate they are equally likely and they follow uniform distribution. So any number can come again. So therefore all following uniform distribution therefore you have to consider only this type of interval say but here you are continuous case now. So this range since you belong to 0 to 1 ln of 1 minus u ln of 1 minus u is nothing but ln of u. So any one you can select, any one you can select between the two because ultimately u belongs to 0 and so ln of u and ln of 1 minus u remains same. So this is you can write or x equals to minus 1 by lambda ln of u. So look at the two scheme. So let me put a color you will get to know look at the two scheme now. So this is one scheme or anyone, anyone both are same now because ln of u and ln of 1 minus u are same because u belongs to 0 to 1 so therefore you can consider either of the scheme. So this is what nothing but here you can see x equals to minus 1 by lambda ln of u. So this is the scheme. Now what you do? You generate any number from the bucket look at here. Now let me keep the say you know linkage now. Suppose whatever you can consider to the bucket now. So what do you do? Or even you can do this whatever. Now what do you do? You go to you know your data set. Now you generate any number say in your interval say let me write you can see here say look at here effectively. Suppose you know this is your say x and this is your capital FX right range 0 to 1 and suppose you have created your cumulative value. Suppose whatever you have created your cumulative value like this say 1 and what happened here initially you are generating random number interval 0 to 1 say you have generated 0.5 say you have generated 0.5 so the corresponding x will come here corresponding x will go to this u generated here look at here here you have generated u look at here here you have generated u and corresponding x you will get here u you generate here u you generate here in this y axis and you will get corresponding x this x will go to your you know system of simulation whether manufacturing system supply chain whatever you are working or you know demand planning you can put it here this x will go. But effectively you are not directly giving x you are giving actually putting the random interval in this inverse scheme and throw that you are generating x. Now suppose next iteration you have generated say 0.4 suppose so 0.4 here say corresponding x so for corresponding that corresponding x will be here if you generate say 0.8 0.8 or 0.8 what about depending on the interval of your data in excel you can do it. So you are getting x this x so different x you are generating effectively not directly but through the scheme of inverse function and this is what the continuous case of distribution how you can generate you know different random number through the inverse scheme and you can get different x and that x is nothing but this in your reverse scheme you have generated and that x in every simulations every iterations of making process you are giving input of x here directly but in implicit manner how by this inverse function. This inverse function can be generated through random interval or random number which is belonging to this capital Fx or cumulative distribution graph or you can say the range of 0 to 1 from there you are generating actually right. So this is what the continuous version of scheme. Now if you think that can we discuss that for normal case also yes so this scheme for exponential case here I have created the scheme or you can see the eulocyte calculation here in the right hand side corner. So this is what the scheme of inverse function this is been already in building excel or in python but here if you see for normal case also in case the data follow normal distribution suppose your x follow normal distribution with mean and sigma set right. Suppose in that case what will be the scheme similarly you can calculate your Fx and then corresponding capital Fx after taking integration of Fx dx you know and then you can assign u equals to this particular formula whatever you will get and then you can inverse calculation you can kill the inverse calculation and then you know you can get to know your scheme x equals to some formula in terms of u and then you can generate same way you can calculate from your graph and you know and you can generate different u here and between 0 to 1 0 to 1 say and then you can get corresponding x corresponding x here x here right. So this is what the scheme inverse scheme but for normal case you know it is very difficult with scheme this density function is very difficult to calculate and the corresponding Fx will be more complicated. So this elastation are been done in the excel or in python say I will show in excel today this function normal inverse function you can use with the data and you can generate the simulation input random number and the corresponding x value the x value the x may be demand x may be planning whatever you can consider x right. Now suppose let us consider the data who follow normal distribution with mean 100 and standard deviation say 20 suppose this will illustrate in excel now for normal case suppose we are going to illustrate now which follow the parameter it can be any parameter right it can follow suppose it is following normal distribution and how to create the entire scheme and generate the number for your in a practical problem let us illustrate that here you can see suppose x is following suppose x is demand say and it is following normal distribution as I talked in the PPT with mean say 100 and standard deviation say 20 suppose you have the data and from that data you have calculated the mean and standard deviation suppose here now we will take that information mean and standard deviation and we will generate the inverse function here and we will calculate say Fx suppose Fx we are calculating here suppose function value function could be supply chain it could be say anything suppose this function formula or nothing but say suppose I am writing say 2x plus say 30 say suppose for the sake of illustration 2x plus 30 this functions is a suppose profit function or supply chain function or a manufacturing cost whatever that you will have to calculate for different data sets of demand right different data sets of demand and data sets are here but you have taken the mean here calculation from the data and suppose standard deviation here but how will generate different this function value for a given demand since demand is uncertain and risky parameter and which is following normal distribution you do not know in a next event what will be the demand since demand is uncertain therefore your you know say supply chain cost or manufacturing cost will also be uncertain or profit will be uncertain here. So, this function will also become uncertain but how to generate that because you cannot take the expected value expected value does not give except mean and standard deviation on an average expectation you can calculate but here we are going to see the diversified data sets through different random number generation and we will see what kind of instances we are getting of demand functions or say you know output objective functions here. So, let us see how we can create this inverse scheme through Monte Carlo simulation first step we have created a RAND function. Now let us see how this RAND function are being generated and the normal distribution scheme can be executed through this Monte Carlo simulation scheme. So, here I will show you how the RAND function are being generated let me delete this first call the RAND function here you can see the RAND function just select it and drag it say for say 20 samples say so you can see 20 sample of random number we have generated like we have generated in the bucket with the example of simulation scheme same way you can generate here right you can you know reduce the number say up to 2 digit or say 3 digit here I have not done that you can do it this range could be 0 to 1 say you know you can mention that 0 less than 1 say. So, it will lie between 0 to 1 and then you can generate the RAND function say RAND number here say and then the normal distribution scheme the inverse scheme how to create it the scheme let me write the scheme the scheme you can write like let me delete it also just call the normal inverse look at here normal inverse select the probability that is a RAND function here and the mean mean of the data we have considered 100 say put dollar sign so that you can drag it and then select your standard deviation you also freeze this. So, we have created the normal inverse scheme here for this data set say and we found the value of normal distribution as x as input to your system the output systems say suppose you have a data set here say suppose you have a data sets here. So, these data sets say you are actually calling here through this distribution scheme normal distribution scheme and this mean is nothing but the average of this data average of this data is nothing but your mean and standard deviation you have calculated here suppose and then the scheme you are generating here in the inverse function look at normal inverse function through this RAND function. So, rather than taking average you are generating through the scheme normal distribution scheme you are through inverse function you are generating it and here you can see so function output say as I mentioned suppose you know you have a system say system and say let me write down here it is as a system say rather than function I can write system output say right system output. So, say suppose 2x plus 30 so what is that here actually this is your system output say system and the output will be here and you are giving input this input will come through inverse scheme. So, first you are generating RAND function your data sets are here your data sets are here data sets are that data sets you have taken the mean and standard deviation you have calculated and through RAND function you are generating that data sets through here like the way I have told you suppose you know your distribution is like this, but you have created your like you know cumulative inverse function right suppose 0 to 1. So, right this is called RAND function you are generating RAND function here between 0 to 1 and then through inverse function you are generating x this is called your x or input this input this is your input and you are getting output of this system say this we are going to show you say suppose now. So, here output will be say 2 into x what is the x this value this x axis say this normal inverse input data that means say your demand sales whatever and then plus 30 system output it is a say objective function it could be say you know profit supply chain cost you know manufacturing process productivity or you know financial statement whatever you calculate it if you drag it for 20 samples say you got the system output. So, this is what the inverse function scheme that means or entire Monte Carlo simulation scheme. So, first from the data sets you get your mean standard deviation and the pattern of the data through the software I will show you how to create this you know data pattern how to generate the data pattern also what type of distribution it is following we are assuming it suppose it is following normal distribution then through a random function you generate your random function I mean random number sample between 0 to 1 and then through this inverse function scheme you get your x input or say the parameter demand sales that input through this you know say column number h and that will go as a input your system input and then you are getting system output through different simulation through different iterations here I have shown you 20 samples. If you have a thousands of sample generation and then you can take the mean of the data system output the confidence interval and the standard deviation the skewness cartosis lower bound upper bound all this analysis statistical inferences you can draw and you can get in in sites of it. So, this is what the basic scheme Monte Carlo for normal distribution. Now, what we will do we will go for different application through the software version. Now, let us see one application with some real life problem where we will use the continuous case of simulation. So, just read this particular example here here we have considered a large catalog merchandiser is planning to have a special furniture promotion for the year and now the company has given a order they have placed a order and initially based on the previous assumption or previous information of the data they have given an order to the manufacturer for 3000 chairs. So, this is the initial order they have given with a price of 175 rupees per unit and companies planning to all are like you know initial assumption initial plan they are thinking all are in expectation mode what will happen in future though do not know, but before the business starts say maybe the winter season many people starts with woolen material selling right they procure from the manufacturer and they sell for that particular time period. So, two months three months the one months of winter over that product will be sold out with some discount price, but during this two three months there will be a peak period of sale. So, this type of problem you can assume it even summer also it can be considered as a similar problem say suppose. So, therefore, they are planning that the on an average per product selling price they will set as 250 rupees per unit. So, that means, they will be having around you know 250 minus 170 or almost 75 rupees profit per chair say you can consider with a woolen materials or anything whatever you think the promotion will last for eight weeks say two months say and then after that the remaining if there is an remaining inventory. So, they will remaining inventory or the excess inventory will be sold out with a discount price of 125 from the current like you know selling price. So, it will be almost 50 percent discount set. Now, the company believe that on an average 2000 units will be sold out during the first eight weeks there might be more than that, but initially they are assuming that that minimum 2000 units of you know inventory or demand will be there sales will be there during the peak period say eight weeks or say two months attentively and if there is a excess inventory that excess inventory will be sold out with a discount price of 125 what would be the profit for the retailer. So, let us see how we can calculate it the basic problem we have not incorporated the simulation Monte Carlo simulation here and the risk analytics and the future forecast. Now, let us see the simple calculations. Suppose P is the profit we have noted it as a profit and you know look at the notation C is the you know initial purchase price R is the you know selling price this is the total inventory order and B is the total demand for that particular period and if you take this calculations here is the total profit nothing but the you know the revenue minus cost the profit R minus C you can see into your demand this demand and then the remaining inventory if there is a remaining inventory what is remaining inventory S minus V S minus V is the remaining inventory you can see these two S minus V is the remaining inventory multiply if there is excess inventory multiplied by the discount price. So, you know what is that R by 2 like you know 250 by 2 minus C C is nothing but your 175. So, this is your total basic problem, but we will understand how Monte Carlo simulation can help you in taking a decision in practical dynamic situation and if you calculate all this with this data will get your profit is exactly 1 lakh right simple calculations I will show you in excel also. Now, in case if you think that look at the problem statement and come back to the overall information that they are assuming that they are given order of 3000 fine, but they are assuming that on an average let me open a highlight point you will get to know on an average they are assuming that you know 2000 will be the demand right during the peak period during the peak period of the season 2000 will be the demand and also they are assuming that the selling price suppose look at the practical dynamic situation what happens in practice. Outskirt of your city or the town you may see that during winter season many people come and they set up their store shop and then they sell the woolen products say you know for say 2 months 3 months. So, think that type of problem suppose for the sake of you know it is a news vendor problem we call it, but just think about the sales promotion problem. Now, they have set a selling price of 250 on an average, but you know they can bargain with the customer say. So, that you know this could have a flexibility. So, if the some customers are very keen to buy the product or they like the product they can sell it by 300 rupees also. They can increase the selling price and if somebody is bargaining and there is no customer on a particular day suppose they can reduce the price with say 200 rupees. So, the selling price can also have a flexibility in practice and also demand based on the past data you have calculated the tentative demand rate on an average, but in practice you do not know demand could be anything it can follow normal distribution it can follow triangular distribution what kind of frequency you do not know right and future actual what will be the demand you know. So, suppose demand is also uncertain in that case what would be your expected profit in that case your profit will not be the fixed one lakh it will be expected profit. So, what would be the expected profit that we are going to discuss now. So, look at this is the general case complete deterministic situation now. Now, we are going to incorporate the uncertainty or the risk parameter into the problem now. Now, we are considering that demand follow triangular distribution right. So, this is first assumption we are considering and then we are assuming that the sales price that the sales price that we have captured sales price follow uniform distribution right. So, with a minimal point say 250 and demand also 2000. So, this is what the information say 2000 as the middle point of triangular distribution of demand we are considering and selling price also we are now considering as a uniform distribution with a variation of say you know say 202 say 300 say it can have a variation. Suppose the retailer can play with the customer say dynamic situation the practical situation we are trying to capture in that case how we can do this using simulations or how the Monte Carlo simulation can be incorporated and you can get the expected profit of your problem this profit will be expected now expected profit now because it cannot be fixed now because demand and selling price are changing in practical cases. So, your profit will be all in a expected mode with standard deviation and confidence interval will calculate the confidence interval also of the data. So, demand is following triangular distribution with a range of like this and that and also say you know and selling price is also following a uniform distribution. So, in that case what would be your overall analysis through simulation let us see we will discuss that using Excel. So, here we have the basic data sets look at the same what I have shown you in PPT here in Excel also same calculation you can see the calculation as it is right 1000. Now, we are putting this distribution to distribution into the data we are incorporating. So, demand is following a triangular distribution for the sake of you can change the distribution based on the previous data patterns based on the historical data and accordingly you can feed the distribution I will show you that also through the software later stage. So, now these two distribution we are considering for the sake of illustration of the model now let us come to the simulation model. So, here if you see manually I have done then I will go to the software and show you the how simulation can be done immediately and thousands of instances you can generate and you can take inferences through the data and make a better decision making for the particular example or in real life situations. So, let us see here. So, here we have considered demand is following triangular distribution here we have using grand function look at the function here we have followed the formula of triangular distribution then we have considered the selling price is also following some you know uniform distribution you can see the formula of lower bound plus the gap into the grand function. So, you can generate different type of initial price starting from 200 to up to 300 because grand function is lying between 0 to 1. So, accordingly at the interval value can go up to highest and to the lowest if it is 0 then it will be come out with the lower point if it is highest it will go to the upper point. So, in between if you generate random number you will get different initial price. So, this is what is our data and this is our profit now look at the for this particular we have through random number we are generating demand and initial selling price now demand is not more no more to 2000 now it is 2461 say and selling price is 251 say now your profit is 1 lakh from 1 lakh it has gone to 162, but for one instances one simulations now if you generate again suppose another demand look at and say another selling price by generating different simulation or you know different input data for a demand and selling initial price you will be able to get different profit here for the sake of illustration I have generated 20 such sample by repeating the process of simulation. So, here I have generated say 20 replications I have done you can see here. So, one simulations I have got another simulations if you change the selling price or initial value you may get another you know say here you will get another value say now look at here it is changing now. So, this way you may get different output. So, here I have replicated the 20 instances the simulation process and the corresponding profit price everything I have calculated here look at profit can be negative also in terms of scale value I have kept here not the actual value there here I have to have put the data direct value, but here I have taken the 3 digit values here and then if you can copy this value here because simulations will generate different data sets suppose I have copied and pasted here as a value and if you take your total observation sample mean of the data look at sample mean of the profit and the demand whatever suppose profit we are counting and then stamp sample standard deviation and then you know the with the error the standard error look at the standard error I have calculated here and then you know that with the range of say 90 percent confidence interval and if you take the lower bound or upper bound using the formula you get look at 90 percent confidence interval we found the lower bound of profit will be 71000 and upper bound could be 90 percent confidence 16168000. So, from 1 lakh it is going to 168000 90 percent confidence if you change the confidence level you will get another interval also. So, you are getting mean value also of your data look at mean value of your data based on this 20 sample if increase the sample size it will change because you are you have a new demand and new selling price you are changing that and you are taking different combinations and you can find you know based on this 20 sample you may get 1 lakh 20,000 as the expected profit mean value expected profit with standard deviation and the confidence interval, but if you increase the sample size you might get your own and average the mean value could be 1 lakh, but variation will be there and the confidence interval will be you will be able to calculate. This will give you much more flexibility in understanding the practical situations what could happen in future and how can you take a recourse action that analysis we can do also. This is a manual calculation through excel of simulations. Now, let us see the same problem using the address software. More detail of the software I will discuss in a next session detail with couple of case illustration also, but let us understand how this simulations are being incorporated in excel and how this particular problem can be addressed. So, I have already installed the address software here. So, you can also install it 14-day trial version are available through you know polyshade or link arrow website. So, from there you can install this 14-day trial version of software and you can practice, but if you want to buy this particular software you have to you know procure through your institutes and through license versions which is very costly, but you can try by in case you like this software. But anyway let us focus here. So, suppose I have installed this software for 14-day trial version say and then we will run this particular model using simulation now. Let us see what could be output, but here you can just come back here the analysis that we have done based on this input data of demand and stochastic parameter of demand and say you know initial price which are random in nature. And therefore, we have generated the simulations and 20 replications we have done it here manually, but here we will do through the software. We will not generate 20 samples we might generate 2000 samples of simulation mimicking process that in the last class I have discussed detail about the basic simulations that part we are going to do it here now. So, the software is here now, but how to incorporate this demand as a here we can see the new formula risk triangle how we can define it let me show you here. Suppose here what you do let me delete it first look at no sale is there now it was 2000. So, I will incorporate and it was 250 say let me delete it and I will start from scratch. So, select it and go to the software define distribution look at so we consider it as a triangular distribution. So, define distribution and suppose this is the range and what was the range we have assumed we can come here let me check again the range I think it was 500 least case and it was I think 3000 I will check and come back again I think 3200 something suppose I will modify it suppose this data were given here. So, let me see here I can show you then so around 5000 and here it is almost say you know almost 3500 sorry 500 to 3500. So, you have set it now you can check again the distribution function look at 500 you have the flexibility to change the data here also and 500 to 3500 with the middle value most likely value like triangular distribution we are considering is a triangular distribution based on the previous data. If you have a sample data and you do not know which distribution it is following through the software you can you know set the distribution also based on the recommendation of that software I will show you in next class in detail of it. Now, we are assuming it as a triangular distribution and now initial price I have reset now I will consider it as a say uniform distribution that is what we have assumed here. So, what I will do I will go to define distribution again and then then we will consider it as uniform distribution is only for illustration only and in the lower bound we are setting it as a 200 the bargaining price with the customer and it can go up to say 300. This is what we have assumed the range of the data for the selling price earlier it was 250 fixed now you are making a flexibility that you do not know you will be able to convince the customer about your product and you can sell it. So, now both the data are random now. So, we have considered them as a risk parameter and any every simulations one input of demand will be generated one input of demand will be generated here sorry one input of demand will be generated here and one input of selling price will be generated and if you consider them and if you recalculate your profit you will get new profits. Now, look at the output cell here you can see the risk output that you have to do, but let me do it again. So, that you should understand how to do it just click it put a backspace this is a simple calculation right it was there here also you can see the original model you can see simple calculation look at here same calculation come back to the risk model and then the output cell for the software you have to click you have to mention that it is the output cell just click that cell and just put output option that is it done. So, now suppose this is your profit right you can put your profit your company profit whatever you want you can put in this back at the output cell name right or you can put the date also whatever anyway. So, this is our profit now it is a simulation is said now now let us go to address and I told you the in the excel manually we have run say 20 samples say now you can run 2000 sample also here look at if you want you can change the sample size how many simulations you want to generate or instances you want to generate from your this particular problem of profit calculation through this retailer example right suppose you want to generate 2000 samples say 2000 sample you want to generate profit for different combination of demand and initial and selling price. Remember this is what the simulation Monte Carlo simulation you are doing in every iteration you are generating look at in the previous class I have discussed detail of it in every iteration you will be generating one demand through your end function and the inverse cumulative distribution remember it is a following triangular distribution and then just now I have talked about exponential and normal same logic here I have assumed triangle and triangular distribution and uniform distribution. Similarly you can take the inverse value and the corresponding through excel all the you do not have to do because it is the inbuilt formula has been built by this been said by this particular software right and then you get one output of you know demand and selling price and you put it here you will get some profit again you change your demand and selling price and another simulations you will get another profit this way suppose we will be generating through Monte Carlo simulation process the scheme that I have talked about the four five steps that is been will be repeated here again now just put it say 2000 simulations you want to do now just run the simulation let's see what happens so look at it since it's a trial version for your information I am sharing here academic use only you can see the output now so we are generating 2000 sample now so 2000 sample has been generated here remember if it is a full access version so you know then this particular you know watercolor will not be there but when you download it for 14 days trial versions you will also see the similar watercolor option that is a trial version for academic use only but you can test this particular software for 14 days and you can get to know inside of it I will discuss more detail about this particular software and with case applications with the actual version of the software with couple of case illustration now let us focus about this outcome say so what what we have found here remember here the output data said here look at the mean almost 99,551 because this is what we have considered the general on and ever as expected profit so mean is almost same but in earlier case it was only one instance you have generated but now you have considered demand and selling price as a random parameter right and because of that because uncertainties involved over there your profit will also follow some you know uncertainty and here you have generated 2000 sample of your profit look at here 2000 sample of your profit you have generated here and through that what do you have found you found the mean of your expected profit and also your standard deviation look at the standard deviation here look at standard deviation is how much 1 lakh 1 lakh is the variation for this data in some simulation you may get a high demand and the high selling price also in that case it can generate through the machine through simulation process and then in that case your profit will be too good too high but there might be case where you have a very less demand and less selling price you are selling also with the less amount say 200 rupees and you are also getting very say 500 demand say so in that case your profit will be very less look at the downside case here so what does it mean it means that your profit has a high variation because of simulation because of simulation you are getting different type of you know instances of your profit and you have captured all the 2000 samples here and you have calculated mean do not be afraid about your too much of variation you can come up with your confidence interval right look at the confidence interval mean median standard deviation skewness castle kurtosis everything you found here so here you can see suppose this data you can copy and you can you know you can go and you can you know import this data and you can write in your excel or you know ppt or you know you know what file and you can go for a presentation also in your project report also wherever you want you can use it right for your project purpose and now look at the particular analysis here suppose here I have mentioned that in the in the manually I have shown you 90 percent confidence interval now suppose here you can see also 90 percent confidence interval the data range etc for this particular data sample now if you want to change that what would be the chance that you know my profit will be say minimum 2 lakh suppose look at what do the chance that my profit will be minimum 2 lakh look at there is 17 percent chance 17.6 percent chance that your profit will be above 2 lakh only 17 percent chance and also what is the chance that my profit will not be go below 0 there is a 19 percent chance that for this variation in demand and selling price that your profit may go below 0 profit may be negative also so make sure of that so you have a confidence interval you can play with that also suppose you know if you want to put say another confidence it will say say 5 percent in the upper side and 5 percent in the lower side you can put that also and you can see what would be your changes in the profit etc so all flexibility are there in your hand and you can make the policy and you can make a statistical decision making also here now you see this profit is been explained by demand and say selling price right but who is making maximum variation in your profit you can see the tonnendograph here because the tonnendograph here you can see if you keep initial price as a selling price here initial price means selling price right if you keep your selling price as static and in that case if you think the demand is uncertain only and in that case what will happen demand is making maximum variation to your profit we call it a tonnendograph but if you keep demand fixed and if you make the variation in initial price selling price only then in that case it is also making variation of your profit but it has a less variation impact in your profit than demand so demand is making maximum variation in your profit so you can trigger demand and if you can calculate a better demand for the next year and you can plan with the more you know data collections with experts opinion of the field and then you can if you can calculate or they make a better forecast of your demand probably variation of your profit will reduce here so this is what the many other analysis also you may able to get from this particular software I will discuss detail about it you can see the spider who is making most maximum variation in that in terms of data so both are making variation but demand is making more variation look at the red color line than your selling price so this kind of analysis you can get to know from the data and you may get a better insights of this simulation modeling through the software of any practical problem this is today we have started with the basic problem of simulation with a software called adrift software more detail about the software we will discuss in the next session and also the features of the software and how to utilize the software for different you know forecasting model look at the time series model we have studied this through Arima model also we will utilize this for this particular software for different other model too so we will see in the next session details of the software how to utilize it but for this particular problem we have solved the Monte Carlo simulation of continuous version manually as well as through the software advanced excel software in the next class we will extend this concept of Monte Carlo simulation through the software called adrisk and different type of prediction model or predictive modeling will be analyzed as well as the case applications will be analyzed through this software so with that let us conclude today's session of continuous version of simulation as well as the if you consider the previous session club together the entire concept of Monte Carlo simulation and its application in different practical problem we will study detail about it through the software in the next session thank you