WEBVTT 00:00:01.000 --> 00:00:11.402 Topic 12. Optimization - Maximizing Revenue and Profit. To optimize means to make the best of, and 00:00:11.404 --> 00:00:18.501 if there's one thing people like to make the best of, it's profit. Let's see if we can create a function 00:00:18.503 --> 00:00:27.401 to model profit. We start with cost. Now before we even look at the setup here, we need to talk about 00:00:27.403 --> 00:00:33.603 the word cost. When you go into a store and you see a product you want to buy, like for example a 00:00:33.605 --> 00:00:41.161 CD player, a lot of times you say to the sales clerk "how much does it cost?" Well, what you really mean 00:00:41.163 --> 00:00:46.105 to say is, what is the price? The clerk is not going to tell you what it costs. The clerk probably doesn't 00:00:46.107 --> 00:00:54.405 even know what it costs. Cost means production or manufacturing cost. Price means what the consumer pays 00:00:54.407 --> 00:01:00.100 for it, so we have to be very careful to keep those words straight as we go through this process. 00:01:00.102 --> 00:01:09.701 So back to cost, manufacturing or production cost. The cost function, which we use capital C of x for, 00:01:09.703 --> 00:01:17.701 represents the total cost of producing x units of a product. There are several ways to form cost functions. 00:01:17.703 --> 00:01:27.300 The simplest kind of cost function would be the variable cost, which would mean what it costs for the 00:01:27.302 --> 00:01:36.500 plastic, the chips, the things inside of here, times the number of CD players that would be manufactured. 00:01:36.502 --> 00:01:44.799 Then you have added to that fixed costs. That includes things like personnel, rental, insurance, whatever 00:01:44.801 --> 00:01:53.601 it takes to have the production facility. That's not multiplied by x, the number of units, because it's fixed. 00:01:53.603 --> 00:02:00.598 Now that's the simplest setup for cost, but it is possible that the cost function can be modeled by a 00:02:00.600 --> 00:02:09.801 more complicated function. This first example is an example of the simple setup. Two dollars for the plastic 00:02:09.803 --> 00:02:18.100 and the casing, x the number of units, 4000 dollars for the fixed cost. A more complex model might look 00:02:18.102 --> 00:02:25.900 like this second one. A lot of times this comes from just analyzing data and then getting a function that 00:02:25.902 --> 00:02:33.400 fits the data the best. Different types of applications have different types of cost functions, and so 00:02:33.402 --> 00:02:40.001 that's one of the things of course you have to learn to model. Alright now once you have cost under control, 00:02:40.003 --> 00:02:47.600 the next thing you go to is revenue. Revenue is the money that the business takes in. Now when we talk about 00:02:47.602 --> 00:02:53.501 maximizing revenue and profit, you have to think of yourself as the business rather than the consumer. 00:02:53.503 --> 00:03:01.199 So here revenue, revenue is a good thing. Now there are two sort of pieces to revenue. The first piece is the 00:03:01.201 --> 00:03:09.000 price-demand function. This is another one of those things that's very difficult to model. It's done basically 00:03:09.002 --> 00:03:17.001 by using long term data studies. In many of the problems that we're going to do, this information will be 00:03:17.003 --> 00:03:24.201 given to us. At some point later we will have to come up with these functions ourselves. Now let's look at 00:03:24.203 --> 00:03:31.100 what this is about. It's called the price-demand function, and the demand means the quantity demanded. So you 00:03:31.102 --> 00:03:38.600 think of quantity demanded as x. You think of price as p. Now it has to be lower-case p to keep it separate 00:03:38.602 --> 00:03:47.301 from upper-case P which is profit. So what we have here in this function is the relationship between p, 00:03:47.303 --> 00:03:58.800 the price, and x the quantity. It can be expressed either as p is a function of x or x is a function of p. 00:03:58.802 --> 00:04:07.502 Most of the time it's more useful when you're modeling revenue to set up the revenue where you have the 00:04:07.504 --> 00:04:17.000 quantity as the independent variable and then the price, p, a result of the quantity. This is sort of like 00:04:17.002 --> 00:04:24.500 what a standard price-demand equation would look like. A lot of times the x is even dropped here, and it 00:04:24.502 --> 00:04:36.700 just says p equals 250 dollars minus 2x. Now that gets put into the revenue function in the following way. 00:04:36.702 --> 00:04:47.199 R of x, capital R of x, revenue, and this is the total revenue generated by producing and selling x units at 00:04:47.201 --> 00:04:54.600 price p. Now if you stop and think about it, it's pretty easy to see where this kind of function comes from. 00:04:54.602 --> 00:05:03.600 You model this after exactly what happens. Revenue is the price that you pay, that a consumer pays for a unit, 00:05:03.602 --> 00:05:13.400 times the number of units that are sold. Now this price is often modeled by this price-demand function, and 00:05:13.402 --> 00:05:21.601 then it's multiplied by x, the quantity sold. So your total revenue function might look something like this. 00:05:21.603 --> 00:05:30.701 This would be the price-demand and then x the quantity. Alright, so now we go to profit. The model for profit 00:05:30.703 --> 00:05:39.202 looks like this. The profit is what's generated, the income generated. It's the part you get to keep after 00:05:39.204 --> 00:05:46.801 producing and selling x units of a product. The setup here is that profit is the revenue that comes in minus 00:05:46.803 --> 00:05:54.100 the cost that it takes to produce the product. So putting together what you saw before, a profit function 00:05:54.102 --> 00:06:03.798 might look like this. Price times quantity, so this part is revenue, and then minus, and then this would be 00:06:03.800 --> 00:06:15.501 a simple cost function, 2x plus 4000 dollars fixed cost. So here we have a model for profit. Now let's take 00:06:15.503 --> 00:06:24.800 a look at some examples of how this would actually play out. One more time though before we leave this, 00:06:24.802 --> 00:06:36.701 profit is revenue minus cost. So here the profit function is the price times the number of units minus, 00:06:36.703 --> 00:06:43.801 and then this is the cost function, which is cost per unit times the number of units plus the fixed cost. 00:06:43.803 --> 00:06:49.501 Now this is just one model, but it is one of the more basic models and it will help us to get started. 00:06:49.503 --> 00:07:02.000 Alright, let's look at an example. In order to sell x portable CD players, a company has determined that 00:07:02.002 --> 00:07:10.701 the price in dollars must be p of x equals 250 minus 2x. Now this in this problem is going to be just 00:07:10.703 --> 00:07:18.501 handed to us. This is the relationship between the quantity sold and the price of the CD player. Each CD 00:07:18.503 --> 00:07:26.798 player costs 2 dollars to produce, fixed cost 4000 dollars a month. Alright, so what's the question? 00:07:26.800 --> 00:07:35.700 The question is how many of these CD players should the company produce and sell per month to maximize profit, 00:07:35.702 --> 00:07:47.598 and what price should they charge? Alright, let's see what we can do with this. In order to find the 00:07:47.600 --> 00:07:55.600 solution to this problem, we go through a procedure that has four basic sort of steps. The first step is to 00:07:55.602 --> 00:08:00.900 create a model. It's to take these words and come up with a function that looks like f of x, except this time 00:08:00.902 --> 00:08:08.900 it's going to be capital P of x for profit. Then the next thing we do is we look for, well we're looking for 00:08:08.902 --> 00:08:16.598 maximum profit, so we're looking for the absolute maximum value of the function over, and we're going to have 00:08:16.600 --> 00:08:22.901 to establish a closed interval. So the second step is going to be to find a critical value in the interval. 00:08:22.903 --> 00:08:29.502 Hopefully there is one. Then the third step is one that you can't skip. You have to be sure that you got a 00:08:29.504 --> 00:08:36.298 maximum rather than a minimum. So you actually check using the second derivative test for relative extrema. 00:08:36.300 --> 00:08:40.502 Except in this case we're going to have a closed interval, so it's actually going to be an absolute over 00:08:40.504 --> 00:08:45.501 the closed interval. Then the last thing you have to do is what you do in any word problem. You go back and 00:08:45.503 --> 00:08:54.300 you read the problem again, and you answer the specific questions. This one says how many players and 00:08:54.302 --> 00:09:09.801 what price? Ok, so let's get started. This is example one. First of all, what are we trying to do? 00:09:09.803 --> 00:09:16.900 Minimize cost, maximize revenue, maximize profit? Go back and look at the problem. In order to maximize profit. 00:09:16.902 --> 00:09:28.800 So the first thing that I always do is write down max profit, just in case somewhere you lose sight of what it 00:09:28.802 --> 00:09:35.802 is you're supposed to be doing here. Then the next thing you have to identify what your main variable is, and 00:09:35.804 --> 00:09:43.801 in this problem it's handed to you. x is the number of portable CD players, so let's write that down. 00:09:43.803 --> 00:10:03.100 x is the number of portable CD players. Ok, now we need to take this model that we've been talking about 00:10:03.102 --> 00:10:14.801 and turn it into a function that represents profit. Well you know that profit is revenue minus cost. Notice 00:10:14.803 --> 00:10:22.600 that I don't have any x's up here yet. We're just talking about the general concept. Now we begin to use 00:10:22.602 --> 00:10:30.800 the variable x, and x is defined to be the number of portable CD players. So I'm going to model profit 00:10:30.802 --> 00:10:36.901 as a function of x. By the way, when you do these problems your understanding of the concept of function 00:10:36.903 --> 00:10:44.399 just grows here exponentially because you're actually creating your own function here. Revenue, alright 00:10:44.401 --> 00:10:54.100 revenue is going to be the number of CD players sold, x, produced x, times the price which is given. So 00:10:54.102 --> 00:11:05.300 revenue is 250 dollars minus 2 times the number of CD players, now that's the price, times the number sold. 00:11:05.302 --> 00:11:11.995 Alright let's stop a second and talk about why that's revenue one more time. You're selling x players for 00:11:11.997 --> 00:11:21.099 that price. That's the money you as the company gets. Then we subtract what it costs you to produce these. 00:11:21.101 --> 00:11:29.901 It costs 2 dollars for each one, and in this case each one is just x, and then 4000 dollars. So here we have 00:11:29.903 --> 00:11:41.100 2x plus 4000. Ok as we go along I'm going to show you some things that are very common errors. One of them 00:11:41.102 --> 00:11:47.201 right here is to forget to distribute the negative sign. So just always check that step. Ok now the next 00:11:47.203 --> 00:11:54.901 thing we have to think about is, does this model hold over all real numbers? Well obviously the number of 00:11:54.903 --> 00:12:02.000 CD players is not going to be negative, and so what is the limitation or the restriction on the domain. Well 00:12:02.002 --> 00:12:07.701 the smallest number, just think about it, what's the smallest number of these you could sell? 00:12:07.703 --> 00:12:17.700 None. So that means on the left-hand side of the interval, we have 0. Now, there's this interesting little 00:12:17.702 --> 00:12:29.501 relationship here. As the number of units sold goes up, it's multiplied by 2 and subtracted from 250 dollars, 00:12:29.503 --> 00:12:38.300 so the price goes down. So what's the smallest thing the price could be? Well again, zero. So what we have 00:12:38.302 --> 00:12:45.600 to do here is we kind of have to go off to the side a little bit and realize that what happens when this 00:12:45.602 --> 00:12:54.698 price right here is zero. Then the number of units sold has to be what? Alright so what you actually do, you can 00:12:54.700 --> 00:13:01.101 do this mentally or you can just do it kind of off to the side and you'll get used to it. You take the price 00:13:01.103 --> 00:13:13.101 function and set it equal to zero. Again, this is kind of a little aside, and you solve for x. So what do you 00:13:13.103 --> 00:13:21.801 get? Well if you do this visually, you get x is 125. So what you realize is that after you sell 125 of these 00:13:21.803 --> 00:13:29.701 units this model doesn't work anymore because the price is zero. 250 minus 2 times 125 is 0. No more price, 00:13:29.703 --> 00:13:38.001 so the model doesn't work past that point. So this function is a great model, but it only works over the 00:13:38.003 --> 00:13:51.700 closed interval from 0 to 125. Now this is the interval for x, so these are CD players. Ok, so we have a model. 00:13:51.702 --> 00:13:58.300 We need to clean it up a little bit so that we can do something with it. Be very careful here when you do all of 00:13:58.302 --> 00:14:10.901 this distributing and multiplying. You don't want to mess this up. This would be 250x minus 2 x squared. 00:14:10.903 --> 00:14:17.100 Alright, now that takes care of distributing the x into here and into here, and then of course as I earlier 00:14:17.102 --> 00:14:25.201 cautioned you, right here, this is the most common error. Anytime you make an error on a problem like this 00:14:25.203 --> 00:14:29.899 you will usually catch it. Somewhere in the problem you will get something that doesn't make sense, and 00:14:29.901 --> 00:14:35.601 then you can usually go back and fix it. Just keep thinking about what you're doing here. So this is our 00:14:35.603 --> 00:14:42.398 model, and you have just created your first profit function, so you should be proud of yourself. Now let's see 00:14:42.400 --> 00:14:50.701 what we can do with it. We have the model. Now it's calculus time. We want an extremum. So what do we do? 00:14:50.703 --> 00:15:01.300 We take the first derivative and what do we get? Well, we probably could save ourselves a teeny bit of 00:15:01.302 --> 00:15:15.014 work here by doing 250x minus 2x which would be 248x, and the derivative of that would be 248. You don't 00:15:15.016 --> 00:15:21.110 have to do that, but I think it makes the work easier later to combine the like terms. Then the derivative 00:15:21.112 --> 00:15:29.112 of this term would be negative 4x. Alright, let's remember what we're doing here. We've taken the first 00:15:29.114 --> 00:15:34.611 derivative, and we now want to know where the first derivative is either 0 or not defined. Again, this is 00:15:34.613 --> 00:15:39.910 a polynomial function, so we're only going to be concerned about where the first derivative is equal to zero. 00:15:39.912 --> 00:15:45.311 And just for a second, why does that work? Because this is the slope of a tangent line to a curve at a point, 00:15:45.313 --> 00:15:54.112 and that's what you want to know. You see if you have a profit function, this is a parabola that opens this 00:15:54.114 --> 00:16:04.110 way, and that's what we're looking for. Right there. And what is the slope of the tangent line right there? 00:16:04.113 --> 00:16:13.710 It better be zero because that's where this horizontal line is going to be, right at the vertex of this function. 00:16:13.712 --> 00:16:29.510 So we solve for x. So we get 248 equals 4x. x equals 62. Now if you're in an algebra mode, you think, ok, 00:16:29.512 --> 00:16:37.010 x is 62, done, circle the answer, go on. No, no, no, no. That's not good enough. That is a critical value, 00:16:37.012 --> 00:16:42.710 and that's nice, but that doesn't necessarily mean we have found maximum profit. You might have found 00:16:42.712 --> 00:16:52.311 minimum profit. So, we now check. We use the second derivative to do this. Ok, the second derivative of this 00:16:52.313 --> 00:16:59.610 function is negative 4, and we don't even have to sub the 62 into here because p double prime of 62 is 00:16:59.612 --> 00:17:06.011 p double prime of any other x here, which is negative 4, which is less than 0, which means the function is 00:17:06.013 --> 00:17:13.911 concave down, which is exactly what we wanted it to be. It is concave down. It looks like that, and so we 00:17:13.913 --> 00:17:23.110 have achieved an absolute max over a closed interval. Now since there's only one critical value in the 00:17:23.112 --> 00:17:27.310 interval and we know the function looks like this, we don't even have to worry about the endpoints. 00:17:27.312 --> 00:17:35.111 This is sufficient. Alright, are we done? No. We have questions to answer, so we go back. How many of these 00:17:35.113 --> 00:17:41.311 CD players should the company produce and sell to maximize profit? Well we got that right here. This is 00:17:41.313 --> 00:17:55.211 going to be the value of x that produces maximum profit. So we're going to put 62 players, CD players. Then 00:17:55.213 --> 00:18:04.710 the last question is what price should they charge? Ok well, where's the price function? It's way back up here, 00:18:04.712 --> 00:18:14.911 and it's the thing that's multiplied by x in the revenue. The price is 250 dollars minus 2x. So price is 00:18:14.913 --> 00:18:30.010 250 minus 2x where x, the x that achieves maximum profit, is 62. So that goes into there. So I have 250, 00:18:30.012 --> 00:18:41.010 that's dollars, minus 124, and so my price that I should charge for these CD players in order to get 00:18:41.012 --> 00:18:53.310 maximum profit is 126 dollars. Now you might think that we should just charge more or that we should just 00:18:53.312 --> 00:19:00.500 sell more, but that's not exactly what happens. What happens here is as long as our price-demand, which is 00:19:00.502 --> 00:19:09.210 what we base this on, is correct, we would actually have maximum profit if we sold that many, manufactured 00:19:09.212 --> 00:19:16.710 and sold that many, over whatever time period we're measuring this price-demand function, and that's the 00:19:16.712 --> 00:19:25.810 price that we should sell these for. Ok, let's take a look at an example where this is not handed to us. 00:19:25.812 --> 00:19:33.610 This one is shorter, but it illustrates a point, and the point is of course how to come up with the price-demand 00:19:33.612 --> 00:19:44.010 function. It's about cars. Ok, so think about car rental. A car rental agency rents 100 cars a day at a rate of 00:19:44.012 --> 00:19:53.910 10 dollars a day. Now that's a current situation that's happening. For each one dollar increase in the rate, 00:19:53.912 --> 00:20:00.700 the rate goes up a dollar. Every time it goes up a dollar. If it goes up to 11, it happens 1 time. If it goes 00:20:00.702 --> 00:20:06.411 up to 12, that's 2 rate increases. If it goes up to 15 dollars, that's 5 rate increases. Every time that 00:20:06.412 --> 00:20:13.712 happens, their statistics have shown, and you gather this information over time, that 5 fewer cars are rented. 00:20:13.714 --> 00:20:19.910 Now it really helps here if you get into the problem. Ok, so let's look at 5 cars. So what you want to 00:20:19.912 --> 00:20:30.011 think about here is you have these 5 cars. If you raise the rate a dollar, you rent 5 fewer cars. So you 00:20:30.013 --> 00:20:39.310 have this balance that's going on. Every time this happens, 5 fewer are rented. So you get the idea that there's 00:20:39.312 --> 00:20:47.310 this thing, maybe this variable, that every time this happens once you lose 5 of these. So if we raise the 00:20:47.312 --> 00:20:56.012 price 2 dollars think about it, how many fewer cars would be rented? 5 times 2. If we raise the price 3 times 00:20:56.014 --> 00:21:06.600 then it would be 5 times 3. Ok, now we're going to try to create a model for this, and let's go ahead and read 00:21:06.602 --> 00:21:11.311 the question while we're at it. It says at what rate should the cars be rented to produce maximum revenue 00:21:11.313 --> 00:21:24.312 and how many should they rent a day? Alright, let's model. Example two. Remember what we do first? We want to 00:21:24.314 --> 00:21:40.510 maximize revenue. Next step, identify the variable. Ok, the temptation is x is the number of rented cars, 00:21:40.512 --> 00:21:47.308 but that's not what's really happening here. What's really happening here is you having one dollar price 00:21:47.310 --> 00:21:59.310 increases and five car rental decreases. The actual thing that's varying here is the number of one dollar 00:21:59.312 --> 00:22:10.711 increases and the five rental decreases. So the variable is actually going to be the number of one dollar 00:22:10.713 --> 00:22:35.100 rate increases, so x is the number of one dollar rate increases. We're modeling revenue. Revenue as a 00:22:35.102 --> 00:22:46.711 function of x would be the price, or the rental price, times the number of units. Now it's not just as simple 00:22:46.713 --> 00:22:57.710 as x times p this time. This one's going to be the rental price, and this is going to be the number of 00:22:57.710 --> 00:23:10.511 cars rented. We started out renting a hundred cars a day, so that's our beginning. But what happens? 00:23:10.513 --> 00:23:15.611 Every time we have a one dollar rate increase we have five fewer cars rented. So we're going to have to 00:23:15.613 --> 00:23:24.610 subtract from this five cars times the number of rate increases. Now the common error on this problem is 00:23:24.612 --> 00:23:30.011 to forget the 5 right here. If you think about it, it has to be there because it's five fewer every time 00:23:30.013 --> 00:23:36.910 you increase the rate by a dollar. Alright, now let's back up to the rate. The rate was 10 dollars. 00:23:36.912 --> 00:23:44.410 This is dollars, and sometimes it helps to just put little dollar signs and little units in here. 10 dollars, 00:23:44.412 --> 00:23:54.610 and then the increase would be 1 dollar times however many increases there are. Ok, so what you have done 00:23:54.612 --> 00:24:02.610 here is you have produced a model. Now, do we have a restriction? Well yes we do. You can always look at 00:24:02.612 --> 00:24:09.610 the part of a function where something is decreasing. What happens here? Well first of all how many rate 00:24:09.612 --> 00:24:21.410 increases can we have? Well, we kind of have to look at this and see. We can have none. Ok. But how many 00:24:21.412 --> 00:24:28.410 could we have? Well, what would happen if we had 20, if we increase the rate 20 times? That would be ok 00:24:28.412 --> 00:24:34.510 on the rental [rate], but what happens to the number of cars? Well, then that means that the number of 00:24:34.512 --> 00:24:42.408 cars rented is 0. So what we're going to do is we're going to realize that 20 here is the endpoint and 00:24:42.410 --> 00:24:49.610 we're going to say that this function is only defined over the interval from 0 to 20. So when we find 00:24:49.612 --> 00:24:56.011 the critical value it has to be in this interval because, one more time, 20 rate increases causes the 00:24:56.013 --> 00:25:02.411 number of cars rented to be 0. Below that it would be negative. So that doesn't model reality. Ok, so 00:25:02.413 --> 00:25:15.010 what do we have? Revenue is 10 times 100 is 1000, and then negative 5 times 10 would be negative 50x, 00:25:15.012 --> 00:25:26.710 plus 100x. So that would be 50x and then minus 5 x squared, and again if you make a mistake here in multiplying, 00:25:26.712 --> 00:25:31.911 you'll see at some point that you get something that doesn't make sense. Let's check this again. It would 00:25:31.913 --> 00:25:41.413 be 100x minus 50x and then minus 5 x squared. Ok. That's revenue. What do we want to do with it? We want to 00:25:41.415 --> 00:25:48.714 find to find the rate and how many cars will be rented. So how are we going to do that? We're looking for 00:25:48.716 --> 00:25:56.115 a critical value, so we take the first derivative. Now this might seem quite amazing, but I've seen it happen 00:25:56.117 --> 00:26:02.612 lots and lots of times. In a calculus class, a student is so busy focusing on the model they actually forget 00:26:02.614 --> 00:26:08.514 to take the derivative. You want to always remember that you are maximizing here which requires you to 00:26:08.516 --> 00:26:19.310 take the derivative and then find a critical value. So we get 50 minus 10x. We set this equal to 0. 00:26:19.310 --> 00:26:29.313 We solve for x. We get 50 is 10x. x is 5. Ok we're not done. We need to know if this is a max or a min. 00:26:29.313 --> 00:26:38.310 We're going to use the second derivative test, R double prime. We get negative 10. So R double prime of 5 00:26:38.312 --> 00:26:43.414 is also going to be negative 10, which is less than 0, which means the function is concave down, and 00:26:43.416 --> 00:26:51.416 yes indeed we found a max over this interval. Another way you can tell is if you look at this revenue 00:26:51.418 --> 00:26:57.013 function here, it's a quadratic function leading coefficient here of, when it's in descending order, 00:26:57.015 --> 00:27:03.029 negative 5. We know it's a parabola, but you do have to be sure that this is in this interval. 00:27:03.031 --> 00:27:07.511 The last thing you do is answer the question. At what rate should the cars be rented? 00:27:07.513 --> 00:27:15.615 Well the answer is not 5. 5 is the number of rate increases. This is the rental rate, so the rental rate 00:27:15.617 --> 00:27:25.411 is actually going to be 10 dollars plus this 5. So the rental rate should be 15 dollars per car. 00:27:25.413 --> 00:27:34.613 So we use this 5 in here. And then, how many cars are rented? Well it's 100 minus 5 fewer for each of 00:27:34.615 --> 00:27:46.610 these 5 rent increases. So that's 100 minus 5 times 5 or 25, so we end up with 75 cars. Let's go back and look. 00:27:46.612 --> 00:27:54.610 According to this model, the rental company would be much better off rather than trying to rent 100 cars 00:27:54.612 --> 00:28:03.410 at 10 dollars, their revenue would be higher if they actually only rented 75 cars at 15 dollars.