Next question is from Anna Schott. How much do you think about your branding or marketing in order to appeal to both men and women? Oh, this is a fun one that we wrestle with our marketing team and have. Big wrestle. And we have for what, three years now, this was a really hard thing for us to accept. And what I mean by that is, you know, when we hired our marketing team, this was one of the first things they wanted to do was, okay, we have to, you know, market the maps program to women, then we need to market it to men. And so let's change the colors of it. Let's do this. And we're like, oh, it was like nails on the chalkboard for us. And I've come full circle on this and have a greater understanding of what this is like. And let me explain how it makes sense to me now and why I'm a little more receptive and accepting to allowing our marketing to go after and target like women specifically, even though I know that, okay, that work, that program would also work for a man. Same workout. Right, same workout for a man. It reminds me of when you, and I know you guys can relate to this. When you got the client and they come in and they spent a lot of, they invest in you for six months plus and they come in and they tell you what they want. I want you to train me six days a week, Adam. I don't want to do any of these exercises. I'm going to keep my Sunday fund, I don't want to cut out this. And I don't want to do any gimmicky. And they give you like all these things.  Eventually I'm going to get rid of that. Eventually I'm not going to allow them to train that much eventually. But I also know that I'm in a service business. This person just met me. This is what they're here for. This is what they're spending their money. So I got to kind of give them what they want and say, okay, and then after I've got their commitment, now I have to build trust with them with my knowledge and experience and value. And then eventually be able to unpack and unravel all these myths that they believe to be true. But I first have to grab their attention or first agree with them just to get them. And then I can start to influence them and explain to them, listen, the same program for you is the same program that I'm going to do for the girl who wants to do this. So it's not a different, but when you search on Google, when a girl across the world who's never heard of Mind Pump gets online and wants to find the perfect workout or the perfect exercise for her, she types it in that way. She types it in as best ab workout for women or best butt exercise for women. Like she searches that way. And if we refuse to find a way to kind of mold to market to them like that, we'll never get that person. I'll never be able to influence her and educate her on how it really works. And so over the last three years, we've had to think a lot about this and find ways that, how can we market outside of Mind Pump very specific to the person that's probably searching a topic all the way down to their sex to what they wanna do as far as their goals and appeal to them to get their attention to then get them to hopefully to listen to the show and allow us to educate them on how it really works. It's all about communication.  Not doing those movements. Old trainer or excuse me, young sal where I first started would have been like, well, I'm sorry, we have to do those things. Here's why, squats are the best. Now you gotta cut that food out because it's how old or more wiser sal says this. Look, we don't have to do anything you don't wanna do. That's it. We don't gotta do anything you don't wanna do. Now I know my goal is to get you to the point where you wanna do those things. That's what a real trainer does because the young novice, not wise sal would lose 20 or 50% of the people. And now I can't help them at all. Now they're doing nothing. They're not doing anything at all. So here's the struggle. The struggle is marketing team says, if you market this program to just women and make it look like it's for just women, you'll sell more programs. That's not gonna work on me. That doesn't work on us because we're gonna look at them and be like, sorry, we're not gonna compromise our integrity just to sell more programs. And then we all sat down and it took us a while and we thought about, wait a minute, let's take a step back. There's a lot of people, now they're gonna go, these people that we're not marketing to the way that they like to be marketed to are gonna go buy some crappy workout program. That's right, they're gonna find somebody else's. That's not gonna work. It's not gonna be effective. They're gonna either hurt themselves or they're gonna slow their metabolism down or all the pitfalls and what's our overarching goal or overarching goal is to truly help people. So why don't we talk to them the way that they wanna be talked to, then when they come in, we train them the right way. So really that's what it boils down to is marketing to specific people is less about, in our case, it's not about fooling people. It's all about, come over here, we'll train you the right way, fine. You want something that says it's just for you? That's what it says, but it's also for these other people too. What do the numbers actually tell us of who's responding, you know? It's like, let's look at that. And like, look, there was a big gap that we had to face that of people who weren't reaching like you're alluding to. It's, and that's troubling to us because that's really our mission is to be able to start that conversation at least. And if I can't start that conversation, then what kind of breakthroughs are ever gonna happen? And it's so frustrating because we know the big leaders in our space and the gimmicks and all these different tactics that they use that are just so dishonest and are just like really just promising the world and not delivering any of the results.  But how do we do this? And so it's, we've had to get really creative and we still do have standards. I mean, there's things we don't use. Like we don't use the whole, you know, switch it out with, you know, the transformation picks and this and that and the other which are very, very powerful tools. So there's other ways to do it with reviews and there's other ways to do it with testimonials where people are like really honestly talking about what we're promoting. So I think that, again, we can use help with this, you know, from everybody that listens in terms of just spreading the word more than anything. It's a fine dance that, you know, we've had to do as far as, you know, keeping our integrity and what's important to us, but then also recognizing that what we're scaling is much bigger than just ourselves and recognizing that not everybody who buys a program from Mind Pump listens to Mind Pump.  And in like to Sal's point, if we don't find a way to market or get to those people, then somebody else will. Somebody else who will appeal to what they want to do and who most likely won't have the knowledge that we have and will give them some shit product that probably won't work for them and they'll stay in that cycle. So, okay, we'll speak to you the way you want to hear information right now to get your attention. And it does, it reminds me exactly what I had to do and that's what got me with our marketing team to finally kind of go like, okay, I understand this because I do remember this was a challenge for me in my early 20s. And when I finally overcame it was exactly what Sal said. I learned to agree, you know, that's like the first rule of overcoming these objections is to agree with them. Like, yeah, yeah, no, if you don't want to do that, we don't have to do any of those things. We'll only do what you want to do. Right, but deep down inside, I know that if I'm gonna really change this person's life, I've got to get them to do all those things, but the way to do it is not to argue with them the first interaction that you have with them, it's to slowly convince them. And that's what the podcast, the email dripping, the YouTube channel, the blogs that we write like crazy, that's the idea is like, we'll use whatever net we can to get them into the circle and then let's start to inundate them with all the really good information and slowly start to convince them to make them want to go, oh, okay. Look, you got, here's the deal. The light bulb has to come on in their own head. Yeah, here's the deal, okay? This is what we're fighting against, okay? Here on one side, the fitness industry monster that says take this pill and lose 30 pounds in 30 days or follow this workout and in two weeks, look like this, right? That's what they're saying. And on this side, we're saying, hey, you're gonna have to make some long-term fundamental changes with your habits, with diet. You're gonna have to work out for a long time and it's a slow process. You got to fix imbalances and work on stability and slowly build strength. And then over time, train your body.  How are we gonna beat them? People want the easy lie. They don't want the hard truth, right? So how do we beat them? We have to outsell them. That's the first step. We gotta do a better job at getting people's attention and selling the truth better than they can sell a lie. You know how hard that is? That is a hard thing. I'll give you, look, I'll give you an example of how, and this is what, by the way, we developed this over years of training clients. This is what you learn as a trainer. If you do this for decades, eventually you learn how to sell the truth better than the other guy can sell the lie. Is if you'll have to learn that, otherwise you'll always fail over and over and over again. At some point you realize, I gotta change the way I'm saying what I'm saying because I'm losing. I'm not winning this battle. I'll give you a good example, okay? Eating in a way to benefit you that makes you healthy, truly healthy. Developing a good relationship with food gives you better health, okay? That's great, sounds real nice. Am I gonna reach the person who's listening right now that just like, I just wanna look good. I wanna look good and I wanna get there fast. I really don't care about that other stuff right now.  No, I'll worry about that later. I just wanna look good. No, I'm not gonna get that person. So how do I get that person? I say this instead, which is also true. By the way, we're not lying. We're always selling the truth. We're just doing it a way that's gonna sell it a little bit more effectively. So here's my example.  If all you do is eat and train to make yourself look good, eventually your health suffers and so do your looks. So now the person who's listening and all they care about is how they look, now their ears perk up. Oh, wait a minute. So if I do the healthy thing, I'm gonna look better. Let me listen to this what this guy has to say. Now I got you. Now I can start to teach you how to do this properly and I've helped you. Otherwise, we're totally screwed and the fitness industry will remain to be a problem of health, not a solution for health. We'll keep fishing.


Hi everyone Joe for jaspyscasebreaks.com coming at you at 2021 top stadium club baseball four box Random team break number nine from a fresh case and the Dodgers in the bottom of the first have the bases loaded with one out against Kevin Gossman of all people and the Dodgers They're second in baseball in Grand Slans with six I think All right. Anyway, big thanks to this group people right here for getting on it. Let's triple you up one spot gets you three teams And all 30 baseball teams are in Let's roll it. Let's randomize it four and two six times one two three four five and Sixth and final time after six we got PJ Down to Michael with the last spot mojo star next to his name 70% of the time last spot mojo hits 100% of the time four and a two six times for the teams two three four five and Sixth and final time twins down to the Cubs All right PJ with the twins buried with the Cardinals PJ with the Rockies and Tigers Paul with the Marlins Michael with the Royals David with the Orioles Kevin you got my Dodgers PJ with the Padres and Phillies Barry with the Mariners PJ with the Rangers David with the Red Sox Barry with the A's PJ with the Pirates David with the Brew Crew Kevin with the Nats Paul K with the Reds Kevin with the Mets Barry with the Blue Jays Michael with the Indians PJ with the Rays Astros and Angels Barry with the Yankees PJ with the Braves Barry with the White Sox Paul with the Diamondbacks PJ with the Giants and Michael with the Chicago Cubs So let's sort by team And I'm gonna flip away from the screen just for a second. I'll come back to it and give you guys a little treaty Trading window here, but let's open up the stadium club case first fresh case And let's see which four boxes we're gonna do Will Smith wanted that Grand Slam All right, let's see on two three four All right, so let's slide Four boxes right here slide another four boxes right there And so when I roll the die Let's go with that one right there when we roll the die We'll go one two three four if I roll a five or six. We'll just roll again one so the first stack all the way over here This put some squiggle marks right there. So, you know that they're from the same case They may be a little mixed up, but That actually just missed ten feet left of the foul pole They've been a grand slam for Chris Taylor. Chris Taylor has four grand slams in his career All right, let's flip back to this screen right here any trades I don't think anyone's really chatting about trades in the chat Maybe we'll just keep the video running trade window going once trade window going twice TWC trade window closed. Let's just get on with it. Let's print and rip. All right two autographs In each box a lot of fun stuff Kevin you want to trade the trade for the Braves? I don't know I don't think PJ is here in the chat Usually checks in usually pretty active in the chat when he's here. All right, so we'll have to move on. There's Jacob de Grom possible NL Sayang front-runner NL Sayang front-runner I would say possibly MVP front-runner Nice Chris Taylor drives one down the right field line bouncing around in the corner. There you go Dodgers I think I two runs on Kevin Gosman Early in the game Two run double Chris Taylor. Do you guys hear that stats Kevin Gosman? Has had 51 at bats where he's had runners in scoring position and they've been Hitters have only hit three off three hits and 51 at bats with runners in scoring position for Kevin Gosman. That is wild There's my Jones. Here we go. Good luck everybody. I remember all carts ship. So if I if I miss something don't worry I'll all get to you. Here's the final printout right here So Paul K with the Marlins gets the Jazz Chisholm a lot of great rookies for the Marlins in there And the Joey Bart rookie card future Buster Posey, maybe PJ with the Giants Kibrian Hayes red foil Nice Pirates that's gonna be also for PJ commercial maybe flip back to the Eastern Conference finals Now Hawks are still cruising 104 to 80 with 420 left in the game Hawks gonna tie up this series Commercial again, that's the worst. Oh This is pretty good. Look at that Daryl Strawberry. Look how young he looks in that picture This looks like a picture from like a family photo album There he is Daryl Strawberry Kevin with the Mets. Is that a catch? Is that gonna be a catch? I don't know. We've got a Ronald Acuna Jr. die cut for PJ and the Braves possibly pre Pre-elicit drug days.  Can they be autographs? Maybe I don't know actually Is that a catch? Dazz camera makes that catch. Absolutely Get those starters out of the game Atlanta. Oh and there's Zach McKinster Let's go back to that Dodgers game because this basketball game is pretty much over someone let me know if the The Bucks make a 30 to 0 run in the next three minutes There's Zach McKinster. Is he in the lineup today? Dodgers Kevin with the Dodgers. There you go Kevin. I think it's a lefty. He should be in the lineup against the righty Kevin Gossum. No, McKinster is not. I don't know if that's a catch for Yelich and looks like It's like something something special is happening. Well, I guess maybe not as special this season, but something potentially Pseudo special happening in Colorado Calderon Jr. Knife shot right there Orioles that's gonna go to David Finals finals a lot of final scores here. Mets meet the Braves 4-3 in Atlanta Red Sox edged out the Royals 7-6 in Boston Blue Jays beat the Mariners 9-3 Yankees beat the Angels 11-5 thing Ohtani had a couple home runs in that game and losing that for Philadelphia Phillies edged out the Marlins 4-3 Nats beat the Rays 4-3 Yes Yeah, the Rockies are about to do something special There's Kenny Lofton. Remember him? His Indians edition of Kenny Lofton Michael Walker last spot mojo strikes again, and that is 44 out of 50 the speedy Kenny Lofton, there's another speedy guy right there, too So we're on average taking taking cuts near the foul line here. Is that what's happening? There's a red foil Jazz Chisholm Nice one for the fish. It'll be for Paul This I like Casey Meis to former number one overall pick Detroit Be for PJ Here's a 6-0 Sanchez. It's pretty good too for For the Marlins again Paul Marlins actually have a sneaky good Sneaky good rotation there young rotation, and there's a Nick Madrigal autograph nice one for the White Sox Barry Roberts Now poor Nick Madrigal. I think he did his hamstring.  He's got his eye how wide his eyes are they're locked in on the ball watching that into the glove Babe Ruth with giant bat always cracks me up halfway through this four-box break next box topper is Aaron judge all rise Barry with the Yankees cool shot. All right. Good luck next box How are we doing on that last filler we got to do for NT down to 24? Hopefully we can get that last filler done by the time I finish these two boxes. We got about another 20 minutes to go here Hopefully we can get that done in the next 20 minutes and we can dive right into NT baseball after you do that last some good show hair Bryce the long locks Trevor Rogers. Whoa Walker Bueller Wow, what a catch Is he okay?  Is he okay? Wow Walker Bueller does does like the tight pants. How does he not rip them? Yeah, I have no idea. Must be a lot of Yeah Oh, there's the auto Davey Garcia. I like this photo That was gonna hit the bottom of the bottom of the wall and be a stand-up double for Walker Bueller All right Yankees that's gonna be for Barry Yeah, he must he must get pants Kevin that has some extra elastic miss elasticity to it You know, it's not just 100% cotton, you know He's got some eggs. He's he's requesting pants that have the extra stretching material There's Justin Verlander die cut PJ with the Astros I'll give Victor Robles the catch on this and these cards being I love stadium club But these cards being flipped around like this From a breaking perspective There's Ryan Mount Castle David other jazz chism I'll give that a catch for Christian Pache Nothing's happening about a Hawks game, right? They win already catch. I'll get that. I'll get that a catch You know, I don't think it's a tops thing to be honest with you Kevin because I think it's at the factory level so I feel like someone a whoever is a running Running stadium club at the factory. I think must have loaded all these up Weird. There's Luis Petino. I don't think it's intentional Luis goes to PJ and the Padres And there's a six o' Sanchez stadium club chrome Another Trevor Rodgers couple Marlins for Paul Paul K. All right final box. We got we made it Autograph maybe box operato That'd be nice But no Boba shed. He's got some good show hair right there, too Berry with the bluebirds Any news on Yanis? What is that? I'm not sure if that's maybe a printing error or or is that a jellyfish and there's the Leody Tavares autograph Rookie Otto for the Rangers PJ with Texas. There you go PJ. I don't know if that's gonna be a catch Tough angle All right, there's Max Scherzer Triumbrance die cut back to the Giants Dodgers game. I saw what happened the oh What's happening in that game where something special is happening? Wow? Kai Tom ended up getting a hit ruining the no-hitter for the Rockies in Colorado, which is a tough It's a tough place to throw a no-hitter They had him no hit through eight innings. Oh, and there's the members only Hassam Kim That's PJ with the Padres. This is this generally falls one per case See the members only stamp right there. What what batter was that? It was the first batter the first batter in the top of the ninth Singles on a sharp line drive to Charlie Blackman and that's a nice start for her Mon Marquez Real nice start got red foil Cole Calhoun and regular Cole Calhoun Both will ship of course and the autograph. Wow. Look at that. There's Dwight Gooden that also looks like a photo pulled from from the old family album here, I like the the sort of like almost a Velour leisure wear tracksuit kind of situation. I like the open jacket there with the chains very 80s Love it. Well, you're looking for autographs Alex and this variations short prints Generally the products like stadium club series one series two. That's what you're looking for Mostly the short prints and just awesome unique photography Right, so there are these red foils Yeah, there's a few case hits to that we can chase a lot of great stuff and it's a pretty pretty budget friendly product too So at the end stadium club is a pretty popular product from year to year and there's more in the store folks check it out I'm Joe for jasky's case breaks.com. Let's do a quick recap. We saw Doc Gooden's buddy earlier, too All right, rookie Otto's like Nick Madrigal is what we're looking for Zach McKinstree and the straw man Darryl Strawberry. Thanks for watching everybody. I'm Joe. I'll see you next time for the next break. Bye. Bye


How do you manage a relationship in medical school? I get that question from you guys all the time. So today we are joined by my beautiful wife and both of us together are gonna help answer just that question. All right guys, what is going on? Luxury for the MD journey, helping you succeed on your medical journey with less stress. And today I have my beautiful wife, Priya. Say hi. Hello. So we are going to answer a few questions on how to manage a relationship because it's hard and we admit that but we've also managed to have a relationship in med school for all four years. We did a long distance. We got engaged in med school and then we also got married at the very end of med school. So we can check out a lot of different boxes. I'm in medical school, Priya is not in med school. And so again, there's a lot of people probably watching this and something that we say is probably gonna fit you. Either you're somebody watching this, wondering how you can have a relationship with somebody in medical school or you're somebody in med school or in residency wanting to know how you can make a relationship work. So first of all, if you enjoy this video, if not for me and then for Priya, make sure you smash that like button and just in case hit it one more time in case it's not working and make sure you consider subscribing to the channel for more weekly videos. So Priya, basically let's do it this way. I'll throw a question out to you and you can give me your perspective and then I'll give you mine from a med student. So I think the most important question is what's the most important thing to make a relationship work in med school? The two C's is what I like to call it, communication and compromise. So having those two. Yeah, so what are some ways that you felt like we kind of communicated or make that work? Well, I mean, in general, just communicating every day. Talking to each other, keeping each other in the loop. What are schedules look like? Because yes, you're busy, but I also do have a schedule too. And so just having that time set aside for each other. Yeah, I think that was one of the biggest things is that as a med student, obviously your life is busy, but somebody outside of med school may not know that. And so the best thing you can do is Priya mentioned just communicating. Hey, like I have a test this Friday, so expect not to hear from me very often. Or like, this is a pretty chill week, let's make some moves to where we can interact with each other. And as I mentioned, like Priya and I did long distance. So we, well, it was like Skype almost every single day our first like year at med school. It was also my first time living alone. So like that kind of support system was really good. So we always had a Skype call every night. Google hangout if that still exists. It does. We would call each other from classes when she would be going from her class and she knew that I would be free. She would just give me a call. So it doesn't be 30 seconds. I'm often interrupted by very last buses. I hate buses now. But those little moments really make it feel like you're not going through the journey alone and the person feels like they're part of your journey. So communication is huge. And number two is compromise. So I think one thing that you hit on was right on the head, which is like, it's also important to make sure that while you're busy you are making them a priority in some form or fashion. I think some of the ways we did that, we interacted with each other almost like every two weeks even though we did a long distance, I would somehow find a way to go back home and drive to Austin, which is where we're both from. In-person interaction. In-person interactions aren't much nicer if you can do them. So I would just drive and surprise her or tell her where I was coming. And sometimes I think it was like for an evening and I'd be out the next day because I'd have a test. But having a dinner together and just having that time is sometimes like worth it. So finding that priority, just like I tell you guys, schedule med school into your life and now you're life into med school. So find out what's important in your life and make sure like those are people and those individuals are blocked off and then find the rest of the time to study. So Priya, next question. What are some good and cheap date ideas to do knowing that med school students are on a budget? I mean, we really, really enjoy the outdoors. So a lot of times we just went on walks around the lake or we went hiking. We're both from Austin originally. So it's very beautiful outside and so it wasn't really hard to find much. That's true. We'd find parks wherever we were and like do a picnic. I think when I brought her down to Dallas once, there's like a nice lake that we found that has like airplanes fly over as they're landing right by the airport. And so we just did like, what was it? Grilled cheese? Yeah, grilled cheese and like some pasta or something. Yeah, so those kinds of dates are really fun.  But look to see what free things are going on. Maybe there's like food trucks or their music events and you can take the person that you're interested in having a relationship with there and it keeps you on a budget but you guys also get to interact with each other. And speaking of nature, that just brings this back to my mind is that for our honeymoon, because we got married a couple of months ago, we went to Banff in Canada, which was beautiful. And so I thought I'd just go ahead and like get a couple clips of B-roll for them. All right, shoot it. All right, so we will do two more questions. I think another big question is just how do you find the time and know if you have the time to make a relationship work in med school? Do you have some things that you feel like maybe we did well that made a relationship work? I mean, again, we talked about the communication and compromise, but I think just it being in a relationship in general, like we weren't always very dependent on each other. That makes sense. So it's like we had our own lives and we were okay with being our own selves. So it wasn't like, oh, I couldn't be myself. Like I needed our relationship to do or anything to do in my life. So I think that was really key. Like just being okay with ourselves before we were okay in our relationship. Man, she just got deep. Hit the like button. No, that's a really good one. I don't think you can really be in a relationship with somebody if you're not okay with just spending time by yourself. Yeah, because the other person's like, you can't rely on them to make you happy. Like you have to be happy yourself. That's a good one. Yeah, before we're getting that. Right, because there are times where I'm not busy and I'm like, hey, where you met student here? Let's hang out and pretty busy. And I have to understand that like she needs to be able to take care of her own life. And if she's busy, then I have to be okay with spending time with myself. Or myself. So like I would go play sports when she's busy and play basketball and she would do things with her friends and her obligations and likewise. And we still do that in residency. I think that actually brings me to the last question. If you could point to end this video is how do you make a marriage now that we're both married?  Yep. I'll show a picture of her ring.  But how do you make a relationship work or a marriage for that fact in residency? Which is even busier. Knowing each other's schedules, like I don't know if you guys probably can't see it behind us but we have a calendar that's put onto our fridge that way we always know each other's schedule. We just need a glance at it real quick so I know when he's on call or when he's on clinic week. Which is almost all the time. Yeah, right now which is his research week so it's actually very, very chill. And for me it's like when I'm going out of town I'm literally hopping on a plane in like 30 minutes. We gotta eat or some food. Yeah, so it's just knowing each other's schedule and knowing when you can have that free time for each other. Yeah, that's a big one. Like for me to communicate my schedule as soon as I know it and I know when we can have dates nights together we went to get tacos yesterday because we both had the afternoon and evening off so things like that are super key. And just living with somebody is a compromise in itself and just finding the things. I think one of the quotes that I heard way back when that's like always stuck with me when it comes to relationships is that but it does have to be important to you that it's important to them. So one of the things for example that Priya hated when I did especially when I was in that school is that would never open. She already knows what I'm about to refer to. I would never open up the packaging all the way just like until I could like get my hand in there and take it out.  And that's exactly how she thought too. But I didn't care. But I did care that she did. So I would just rip it off and I still do that now. Like I read all the rappers. She can be the judge or remote because I know it's important to her. And there are things that annoying me don't bother her at all but she'll do it because she knows that important to me. So finding those areas of compromising how can I make their lives easier so that way our relationship can flow and get better. But I think that's a good place to end, right? So guys, if you guys have more questions about relationships or just making anything in that med school work or if you just want to see Priya in future videos first of all, make sure you smash that like button. If you guys have any more questions make sure you comment down below. And again, if you did enjoy this video I'm gonna say it one more time help this video out and show it to more people looking for help in a relationship by just hitting that like button or sharing it with a friend. But thank you guys so much for watching. Hopefully we've been a little help on your journey today and thanks for joining us on ours. See you guys later. Peace.


There is a video built on Ukraine independence day, where Serkiy Starma is standing on a field and saw through a plane. Behind him, British troops are training Ukrainian soldiers, and you can see tanks in the background, hear military shots from weapons, and it seems like there's nothing in this world that makes Serkiy Starma more proud than to stand on that field, is rallying behind the British ruling class, condemning Russian aggression, and declaring his unshakable support for NATO. Flag-waving patriotic Serkiy Starma takes every opportunity he can to express his love for NATO. He is of the opinion that NATO is a defensive alliance that never provoked any conflict in him. And in this video, he announces that, written together with its NATO allies, are standing united against Russian aggression. He emphasises that the United Kingdom will never be politically divided on these questions. Serkiy Starma proudly stands hand in hand with the Tory government in his support for NATO, who supposedly are fighting for peace, freedom and democracy. Marxists, we are internationalists, we fight for an international socialist revolution that can overthrow the horror of capitalism. Marxists therefore can never be seen to take the side of any ruling class, of any country at any point in time. We stand on the side of the working class all over the world. This is not out of any sentimental reason, but a practical one. Capitalism is an international economic system, so the fight for socialism must be international too. And this may seem like a very obvious principle, but many people within the labour movement, this principle goes straight out the window, especially when war breaks out. And we have seen this many times through history, especially when it comes to the question of NATO. And NATO paints a very pretty picture about themselves, or being non-aggressive, working for democratic values, but if you scratch on the surface, you will quickly realise that this is just a pretty picture. The many NATO wars which have been carried out in the name of human rights, peace, democracy shows the real nature of this alliance. The reality is that NATO is and has always been a brutal weapon of Western imperialism, especially US imperialism. So to understand why the United States needs such a tool, we need to understand imperialism itself. Let me explain that imperialism is the highest and final stage of capitalism. It is based on the development of monopolies, where concentration and centralisation of capital is in the hands of a few capitalists and banks. Imperialism is the naked domination of finance capital. And as capitalism developed, production expanded, the market within the nation's state became too small, and the surplus of capital in the hands of monopolies and banks then began to be exported to other countries with the aim of making super-profit. These capitalist countries then start to dominate the world by dividing up territories and markets. Capitalism is not just an evil idea, it's the logical development of capitalism. The historic role of capitalism was to create the means of production into giant monopolistic firms and to establish an interconnected capitalist world market. And today, with the concentrations of capital in a few key monopolies and financial institutions, the capitalist nation-state is very much interlinked with these private and state monopolies. And instead of having this so-called free market and free competition, you have national states backing monopolies' interests in making profits. For example, companies from the United States and China are dominating finance global 500. Over two thirds of the total brand value in ranking is from these two countries. The US accounts for 49% of these values, China at 19. The United States is the biggest economic power in the world, and obviously America wants to keep it that way, and they have different tools that serve US imperialism. NATO is one of them. Because war is the natural outcome of imperialism and capitalism, let me explain that the course of war in a modern epoch is a division of world-competing imperialist nation-states. And conflict and war will inevitably erupt from the contradictions between these nation-states interests, which at the end of the day is the interest of finance capital. After World War II, the United States became the capitalist superpower on the planet. And one reason for that was that the United States had accumulated super-profits from the war production. And to further dominate the world market, the US pumping huge amounts of money through the famous Marshall Plan into countries like Germany, Italy and France. And they did so out of fear of the advancing Red Army, but also out of fear of revolutions in Europe. The aim was to secure capitalism in Europe and to tie it closer to US imperialism. And the distractions that were caused by war meant that huge reconstruction programme in Europe was necessary, and this laid the basis for the biggest economic boom in history of capitalism. But just like the US, the Soviet Union also came out strengthened out of the war because they had been the main force of beating Nazi Germany. This meant that the entire cause of international relationship was on one hand, dominated by US imperialism and on the other, Soviet bureaucracy. So the US find itself in need of a military alliance to fight the evil Russians. And on NATO's website, you can read the following. In 1949, the primary aim of the treaty was to create a pact of mutual assistance to counter the risk of that Soviet Union would cease to extend its control of Eastern Europe to other parts of the continent. NATO was founded as a weapon of US imperialism to fight the Cold War. It has remained a weapon ever since. Now, of course, it would be outrageous for NATO to openly admit that they are a tool of US imperialism. So it has to be covered up in lies. And one of the main lies is that NATO is committed to the principle of individual liberty, democracy, human rights, the rule of law. And they repeatedly say that NATO is a defensive alliance who are committed to peaceful resolutions of dispute. According to them, NATO has never started a war. Instead, they argue if the chromatic effort fails, it has military power to undertake crisis management operations. So if you were to believe what NATO says about themselves, you would think that they are a peace-loving defensive alliance who's fighting for democracy. They even go so far that they say that they work for the women's cause and are engaged in the fight against climate change. But as I said earlier, you only need to take a quick look at NATO's war interventions or what NATO likes to prefer it to call crisis management operations in Yugoslavia, Afghanistan, Libya to see that their work for democracy is a complete lie. I don't think that any workers in these regions or these countries would agree that NATO bombing helped women there or implemented peace and democracy. The reality shows a complete opposite. Research from Brown University confirms that at least 929,000 people died directly due to the violence in wars led by the US in Afghanistan, Pakistan, Iraq, Syria and Yemen. And many times more have died indirectly to these wars due to the effect of malnutrition, damage infrastructure, environmental integration. So so much for human rights and fighting climate change. These wars reveal the true face of US imperialism and NATO is a brutal killer machine. All these talks about human rights, non-aggression, crisis management operations are just empty words. NATO countries have shown count this time that they are waiting to kill civilians, destroy whole regions to get control over territories. But since the war broke out in Ukraine, the question about NATO and its involvement has come up and in order to defend themselves, NATO is answering what they call myths about NATO. So for example, they say that it's not true that NATO is a war with Russia in Ukraine, that NATO never promised Russia that they would expand up the Cold War. And they say it's just a myth that NATO is an aggressive threat to Russia. But just let's take a look at the role that NATO has played in the war in Ukraine. Because even though they plead to be innocent, it is clear that the West using NATO have been provoking the conflict for their own imperialist influence. From the start, this war has never just been about Ukraine and Russia, but a proxy war about market and military territories between Western allies and Russia. And this has been going on for many years where the Russian Ukrainian ruling class being divided and pulled in different directions. On the one hand, US imperialism and on the other hand, towards Russia. This erupted in 2014 when the pro-Russian Djurankovic regime was overthrown by a movement that was led and that was supported by the US imperialism. And since then, NATO has been trying to keep Ukraine firmly under the influence of US imperialism and not allowing any Russian interference. Since 2014, Ukrainian military has been armed and trained by NATO countries. Earlier this year at the NATO summit in Madrid, it was agreed that they will provide even more military help and support to Ukraine. Between 2014 and 2022, the US military aid to Ukraine amounted roughly to 3.5 billion dollars. And since the war started, Ukraine has been granted more than 13 billion dollars worth of military aid. In NATO countries, they see this as a very well-spent money because it's not because they care about Ukrainian people. US and Western imperialism only spend this money because they see Ukraine as key means of isolating Russia military and economically. The military support in itself won't put an end to the war while the arms industries are looking towards super-profits. The conflict is only being prolonged at the cost of millions of Ukrainians who are seeing their lives and homes being destroyed. But as I've said, NATO completely denies their involvement in the war. Instead, they argue that it was Russia that unprovoked attacks Ukraine. But this in itself doesn't explain anything. To find the root of the conflict, we have to go back to the time before the collapse of the USSR. In the late 70s and early 80s, the Moscow bureaucracy had become a complete better of Soviet Union and it had been disingenuous into crisis for several decades. But by the late 80s certain layers within the bureaucracy and the interbegancia they were moving towards a position towards support of market economy. At the same time, you have also had serious talk about unifications of East and West Germany. And by the late 80s, certain layers between Western capitalists then started to see basically the possibility of having market economy in the region. For them, this was a massive opportunity of crashing the Soviet Union and opening the doors to more profitable markets in the whole Eastern Europe. During this period you had the Soviet Union military alliance, the Warsaw Pact, that had their soldiers stationed out in Europe. And they were there as a guarantee against the Western attack but also to secure the control of the Soviet bureaucracy. And to make sure that the Warsaw Pact did not intervene in the process of capitalism returning to Europe, the Soviet Union were given several promises by Western leaders, but they would not threaten Soviet interest and securities. So Soviet Union leaders, they were led to believe that NATO would not expand in former East Germany or Eastern Europe. On the 1st of January 1990, the West German Foreign Minister, Genscher, said that in a speech in order to not harm Soviet security interests, NATO should rule out expansion of its territory in the East, i.e. moving closer to the Soviet borders. One month later, the US Secretary of State James A. Baker promised Galba shoulder if Soviet Union accepted the new United Germany that they would join NATO. NATO would not expand one inch eastward. A treaty of Germany unification signed by two German republics, Soviet Union, France, United Kingdom, United States, in November 1990 stated that the new United German were free to join NATO, but no foreign troops will be stationed in Germany. Then the following year, just a few months before the Warsaw Pact ended, the John Major told Galba shoulder that we're not talking about strengthening NATO and on the question of NATO expansion he said nothing on that sort will happen. Obviously these promises were being broken a few years later and this goes to show the value of promises made by representatives of the ruling class. Gentleman's agreements are worthless. What really decides is the force on the ground, it's a truth on the ground and Putin, he has learned this from NATO. And with the collapse of the Soviet Union in 1991, Russia became now a playground for oligarchs and western finance capital. With the return of capitalism, Russia faced an even deeper crisis. For example, GDP fell 10% in 1991 and to tackle the crisis, President Jeltsin, who was the man who really opened up for capitalism, he asked West for aid and investment to help carry out his reform program and western finance experts, they estimated that what was needed to take Russia out of the crisis would amount between 76 billion dollars to 167 billion dollars each year for 15 years. But Western imperialism wanted to keep Russia and Eastern Europe economically weak and that plan was never to take Russia and Eastern Europe out of poverty. But to remain a weak region for US and Western imperialism to dominate and in the end, Russia was only granted six billion dollars to stabilise the ruble and a small loan of 24 billion from the IMF and both Jeltsin and Putin relied heavily on the West to maintain their rule. The return of capitalism to this point meant that it was US dominance over Russia. Moscow was being overrun by foreign businessmen and speculators and this is really what imperialism looks like. After the fall of the USSR, US became the superpower in the world. It allowed the US imperialism to intervene in former Soviet spheres of interests and American imperialism taking advantage of to start to see the Balkans, Yugoslavia, Iraq, places that they never dared to touch in the past. But it was not only US imperialism that had an interest in the East. Germany became a key player in the economy of Eastern and Central Europe and the Balkans and they played a decisive role in the reactionary breakup of Yugoslavia. Sweden ramped up their investment in the Baltics. For instance, just two Swedish bank SEB and Swab Bank now own more than half of the banking system in Baltics. Eastern Europe industries began to buy up, no Western Europe industry began to buy up Eastern Europe. For example, the German Volkswagen to Goviskoda. But as the political and economic crisis in Europe, Russia developed, a split between the different wings of the rational rebirth began to occur because the Russian capitalists, they had their own interests which did not necessarily coincide with the West and when the Russian economy began to get back on its feet in the early 2000s, the Western imperialists felt the need of procuring their interests in the region. And in the years to come, the so-called defensive military alliance NATO revealed that they were an aggressive tool of US imperialism. In short order, it began from 1999 to 2004 to incorporate most of the former Warsaw Pact into NATO. The inclusion in particular of the Baltic space brought NATO right up to the border of Russia. And if anyone has ever doubted NATO's aggressiveness, they can take a look at NATO's 78 days bombing of Yugoslavia and Serbia which caused enormous economic damage in human life. NATO not only intervened in a non-NATO country, but took sides against the Serbs which were allies of the Russians and they did so without a United Nations Security Council approval. And this was followed by interventions in Kosovo and in The Independent you can read about NATO's aggressiveness. They say NATO promised to attack only military targets and for as long as they thought they could crack the Serb military, that was what they did. But now they're doing just what the Americans did in Iraq 1991, spreading the war to civilian targets, to bridges, to electricity stations, factories with the excuse that they are also used for the military. The war in Yugoslavia and Kosovo proved some is claiming that NATO never provoked a conflict, it's a complete and utter lie. NATO then continues its program of expansions and these provocations really enraged the military elite in Moscow because under Putin things began to change. He represented a section of the state and the elite which wanted to stop the decline of Russians in the world's relations and demagoguery. He appealed to the feeling of humiliation and hatred that the Russian masses fell towards the West for its conduct after the collapse of the USSR. And at the same time in the early 2000s the American was pushing even further to the near abroad of Russia and tried to incorporate the Ukraine and Georgia into NATO. Both of them bore the rights up to Russia and in the summer that took place in 2008 in Bucharest both Ukraine and Georgia requested to John NATO. And from Moscow's point of view this was one step too far and they did not hesitate to use military power. Russian's army was sent into Georgia and Georgia was swiftly crushed and this intervention really revealed on the one hand the limitations of US imperialism but also Russia's growing confidence and power. So when Ukraine raised their aspirations of joining NATO that did not go very well done with Russians. And let's do a little thought experiment of how the US would react in a stimulus decree. Let's say that China are forming an alliance in defense of peace and justice together with Cuba and Mexico with the aim of having also including Canada. They start to invest millions of dollars in a pro-Chinese party inside Canada. That party led an insurrection that resorted the overflow of Justin Trudeau's government. And this pro-Chinese party put themselves into power. Once in power they began to ban American accents, start oppressing Americans and say that Canada we have now aspirations of joining this Chinese military alliance. Then China start to place weapons station troops permanently, carry out exercises with over 30 000 soldiers right on the border to the US. What do you think would happen? Would Joe Biden just say, let it be, don't worry about it. And countries like Canada are free to join any military alliance if they wishes to do so. Everyone has free will and can do whatever they want. Sovereignty for the Canadians. And you can see basically that this, you would not let this happen. You can see how the US is reacting to foreign countries placing missile closer home. For the US the whole America is a no-go zone while they clear this country. During the Cuban missile crisis the US threatened with nuclear war over the presence of Soviet missiles and troops on Cuba. And this little fault experiment is basically what happened in Ukraine 2014. For years the US have tried to influence politics in Ukraine and make it more friendly to the EU and NATO. The US Assistant Secretary of State admitted having spent five billion dollars to secure independent to get their policies implemented. But in the last decade there has been a relatively decline of US imperialism. And in 2008 you started to see clear symptoms of that. Where for example the economic crisis costful and unsuccessful war in Afghanistan, Libya, Iraq, Syria. So with this relatively decline of US imperialism we have seen countries like Russia and China flexing their missiles. And when 2014 Crimea was joined with Russia following the referendum the US ruling class could do nothing but just raise their anger in words. The United States of course the world still that the world's greatest economic and military power is annual military spending roughly equals the top 10 countries combined. So on the world scale there's no power whatsoever that can challenge the United States. But on a regional level the US can no longer claim to be the strongest power everywhere. And this was reflected in the last NATO summit in Madrid earlier this year where they discussed the question of China. What worries the United States is that the potential dominance of economic dominance of China basically. And they're also worried about Russia now becoming a closer ally to China in order to what they say undercut rule-based international orders. And at the Madrid summit countries like South Korea, Japan, Australia, New Zealand were invited. None of these countries are members of NATO. Inviting them is the provocative move against China.  The reason why NATO is raising this question about China is because it is the main competitor of US imperialism. Last week Joe Biden produced this document called the American Security Strategy Government where he outlined that Cuba is the most serious challenge facing them. Biden said that the US is trying to fight countries like China and Russia by quote investing in American power and influence, building strong coalition to shape the global strategic environment and strengthening the military to ensure it is equipped for an era of strategic competition with major powers. Biden then went on to say that the US would act to ensure that it was out competing China in the technological, economic, political, military, intelligence and global governance domain. And this document is a very good description of what imperialism really is. And as Marxists we can have no illusions whatsoever that NATO or ever give our support to this alliance that we can have no illusion that they will ever bring about peace, democracy and so on. If we did give illusions we will be siding with American and Western imperialism, we will be siding with the Tory government and the British ruling class. The British working class interests has nothing to do with this. They are the ones who cry crocodile tears over Ukraine and Ukrainian migrants. The ruling class and the Tory party, they don't care about Ukrainian whatsoever. Britain together with the US were the ones who were ramping up the conflict in the first place because as you remember the war had a very convenient effect of cutting across the scandals that Boris was facing at home at the time with the party gay scandals. Trapped workers from the problems at home. And now the government is aiming to show that they are the most loyal members of NATO spending more money on the fence. Less trust, she said earlier this summer that the two percent of GDP on the fence should be the floor not the roof. At the same time they turn a blind eye to the bad migrants from Africa and Middle East who are fleeing wars and conflicts caused by Western imperialism and NATO interventions. So to not side with the ruling class should be crystal clear for anyone who say that they are acting on the interests of the working class. But instead we see many examples where there has not been the case. Jake Circus Damer for example who said that to condemn NATO is to condemn the guarantee of democracy and security in France. He even threatens to kick out MPs who has a position against NATO from the party. Now it's not very surprising that this comes from Circus Damer. He's after all an agent of the class. What might have surprised a few people on the left were the fact that social democracies in Finland and Sweden and decided to apply for NATO membership. These countries have for years been holding up this act of being neutral and keeping away from conflicts. Especially Sweden who's been dragging that we have not been in war since 1814. But the truth is that Sweden is not very peaceful as they say they are. While Swedish imperialism is too small to play in a significant role in war interventions their main role has been to export weapons to those wars. Last year Sweden sold most of their weapons to United Emirates who are at the moment killing the Germanite people. Sweden the Swedish imperialists they have their own reason of why they want to join NATO. As they are advancing their financial interest in the Baltics and by formally joining NATO they can have a certain influence over this alliance and defend basically their interests and profits in the region. So with the invasion of Ukraine it has been a massive propaganda campaign to rally Swedes behind the Swedish European and American imperialism. They have deliberately also ramped up they are already ongoing anti-Russian propaganda and claiming that Putin is going to invade Sweden by taking over Gothenland island in the Baltic Sea. This is of course outrageous it won't happen but the real aim is really to scare the workers into a membership of NATO. Earlier this year you had the social democracies. Democratic leaders in Sweden and Finland had a meeting where they discussed the question of NATO membership and present at this meeting was the head of the families in Sweden the main capitalist family in Sweden Joakob Vallemberg who owns SAB which is the biggest Swedish exportable arms. This family also owns the bank SAB which I mentioned before has major interest in the Baltics. So it makes it very clear that under bourgeois democracy it is the finance capital that dictates the politics. The social democratic governments has been studying faithfully behind the Swedish ruling class. They are the party who was leading Sweden into NATO sending arms to Ukraine and increasing their military spending 2% of the GDP and this is of course a complete betrayal by Swedish leaders of the workers movement who ends up defending their own bourgeoisie and Western imperialism. And as you might remember Turkey wanted to use the situation to strike a blow against the Kurdish struggle. In order for Sweden and Finland to join NATO Turkey wants Sweden and Finland to give them full support against the threat of its national security. i.e. they want to have Sweden and Finland giving them full support to Arhugam to brutally oppress the Kurds and this is very ironic that NATO is being portrayed as this alliance against oppressive autocratic ruthless regime in Russia. These same ladies and gentlemen are blissfully contained that one of their own key members in NATO Turkey are in prison torturing and killing the Kurdish people. It is scandalous that these leaders of the labor movement is giving Turkey a hand of oppressing the Kurds in order to join NATO and this shows the upper hypocrisy on the one hand by the labor leaders but also by Western leaders of imperialism and it shows their true aims. We can have no illusion whatsoever that NATO will bring peace, human rights, democracy. You have people like for example NATO defender Paul Mason who calls himself the Marxist but he always ends up on the side of imperialism and the British ruling class and he says we need to strengthen NATO but we need to also make it more democratic so he puts forward a program for us. It is laughable really to think that you can reform the weapon of US imperialism to become a defender of democracy but Marxists don't just say no to NATO our slogan doesn't end with abolish NATO in the last analysis NATO is just a cover for US and Western imperialism. Trump a couple of years ago you might remember he threatened to pull out of NATO would that mean the end of US dominance and Western alliance military? No probably not they would just find another way to defend their interests maybe for United Nations to fight NATO means fighting against imperialism and capitalism and Marxists we are not pacifists we not just abstractly are against war for example you have people like Jeremy Corbyn and Sotna war coalitions who calls for negotiations and diplomatic solutions and saying that war is unnecessary we just move together around the table and come to an agreement but no diplomatic solutions is ever going to be able to stop the real cause and the underlying cause for war which is capitalism as mentioned before these gentlemen's agreements are completely worthless they don't change any of the contradictions of war in the government pacifism calls for inaction and you cannot fight for peace by doing nothing so at the end of the day pacifism ends up defending imperialism for us it will never be the question of taking one side of the ruling class against another the war in Ukraine is a reactionary periodist war on both side one of Russia and one of NATO for Ukraine NATO will never bring peace an imperialist peace will only continue by war of other means in time prepared for future war in the future the real alternative is to struggle against imperialism and to overthrow capitalism and that's why we say no war but class war our starting point must always be the class question this task is to expose the hypocrisy of the ruling class especially our own capitalist class the main enemy is a home the best way to help the working class encounters like Ukraine, Libya, Syria is to take up the struggle against their own bourgeoisie and call the workers of the world to do the same against their own ruling class the struggle for imperialism against the imperialism and capitalism must therefore be based on international working class solidarity and a program for socialist revolution


Hello everyone. So what we will do in today's lecture is we will apply the dynamic programming algorithm to a problem that we have seen so far which is the problem of inventory control. So recall this problem from one of our earlier lectures. The inventory control problem had a finite time horizon n denoted by n. The problem was to decide how much stock or how much inventory to order at the start of every time period. So the time was slaughtered like this from 0 to n and what we wanted to do was how much stock should you be ordering at the beginning of each of these time periods. So the assumption was that the stock that is ordered at the beginning of time period k is immediately available at time period k and but what we do not know is what is going to be the intervening demand during the time period k. So the sequence in which things became known to us that we came to know of the amount of stock we had at time period k. This was denoted by xk. So xk is the stock available at the beginning of time period k. On the basis of that we chose uk which was the amount of stock to order at the beginning of the k time period and the delivery was immediate following which the demand was realized that is wk. So demand during the time period k and we assumed that these w 0 to w n minus 1 were independent. So with and the problem the objective was that we also had assumed that demand that is not fulfilled can be backlogged. So any unfulfilled demand showed up as a negative stock in our any as a negative available stock. So the amount of stock available was given by this linear equation xk plus 1 equals xk plus uk minus wk. The objective was to minimize a total cost. So there were the cost at 3 components. There was a cost there was a terminal cost for being left with inventory xn at time n. There is a penalty r of xk which was the penalty for having excess stock or positive stock xk or in or for holding excess inventory or equivalently for negative stock for shortage of unfulfilled demand. Then there was a cost also of ordering additional inventory this was the purchasing cost. So c that was given in this linear form c times r of c times uk. This is this these 3 together made up our total cost. So if you recall we also had this kind of a flow chart where said the xk is the stock at time k the stock you order at time k is uk that results in a cost r of xk plus c times uk and then the inventory system produces for you the stock at time period k plus 1 by using xk uk and the demand that gets realized in period k which is denoted wk. And the objective was to minimize this total cost which is a expectation of r of xn plus the summation of r of xk plus c times uk. This was the problem. What we will do is now apply the dynamic programming algorithm that we studied in the in our previous lecture. So let us recap that as well. So the dynamic programming algorithm essentially asked us to do the following. It asked us to define a sequence of functions at the nth stage the the function was defined as simply the terminal cost. So jn of xn is is equal to was defined as gn of xn. jk of xk was defined as the minimum value of a certain optimization problem. The minimum was taken over all actions you can take at at at time k as a when you are in state xk and the cost of that optimization was the expectation of the stage wise cost plus the plus the jk of the jk plus 1 of the next of the next state which is which is given in terms of these dynamics here. So this this is therefore the stage wise cost plus jk plus 1 that then defines for you jk and then you recursively define for all these functions from jn all the way down to j0 and the algorithm told you that j star of x0 which is the optimal cost is actually nothing but j0 of x0 where j0 is derived from this from these iterations. The the functions jk were called are are are are called the cost to go or sometimes also called the value function right. So this jk is what is is plays an important role in in all of this in and another object that plays an important role is the minimizer that we find from here. If you look at the minimizer that comes from this optimization problem the minimizer is you let us denote that as uk star this minimizer is a function of xk. So let us write that as uk star equal to mu k star of xk and that then minimizes the right hand side here this particular expression and it minimizes it for each k and each xk remember that we have to find these jk's and jk and jn are these are functions that means they this evaluation has to be done for every value of the state. So it has to be this has to be done for all xk here and this one also has to be defined for all xn okay. So now we will see a demonstration of this for the inventory control problem okay. So let us come come to the inventory control problem let us let us start writing out the dynamic programming algorithm for the inventory control problem. So what does the dynamic programming algorithm ask us to do it says first do first look at stage n and at stage n actually the answer the what you need to do is very simple. So jn of xn should is to be taken as the terminal cost and what is the terminal cost the terminal cost is remember r capital r of xn this is the terminal cost. So jn of xn is equal to the terminal cost which is r of xn. Now if you look at the dynamic programming algorithm now it what it asks you to do is compute jk for each k from 0 to for k bit going from 0 to n minus 1. But how does one do that practically what one has to do is do this step by step for each for each k. So one begins first with k equal to n minus 1. So let me write here k equal to n minus 1 and then write jn so jn minus 1 of xn minus 1 this is equal to now what is this equal to we can look at this from the formula of course but let us just think about it intuitively. So xn minus 1 is the amount of inventory that you have at the beginning of time period n minus 1. So after this that is only this is the last chance for you to take a decision. So at you take a decision at time n minus 1 after that the system will the demand will come in and the system will go to this to state xn at time n at which the problem ends. So the cost that you are going to incur if you are at a if you are in state xn minus 1 at time n minus 1 the what you really have is a tail sub problem of length only 1. So there is a cost that you incur in that problem which is the stage wise cost of stage n minus 1 and a terminal cost of being of whatever which is a function of whatever state you end up with at the end of the problem. So therefore the minimization here is over the sum of these 2 costs which is the stage wise cost in stage n at stage n minus 1 and the terminal cost at stage n. So intuitively this is exactly what you should be doing you should be minimizing the sum of the stage wise cost and the cost that you incur at the end of the problem. And that is precisely what the dynamic programming algorithm also ask you to do. It ask you to minimize you over minimize with respect to u n minus 1 where u n minus 1 is chosen from this set of feasible actions u n minus 1 of x n minus 1 the expectation of what is the expectation over the expectation is over the stage wise cost. Now what is the stage wise cost that you will that you have well the stage wise cost that you would have is you if you order u n inventory worth u n minus 1 the stage the cost that you would incur is equal to c times u n minus 1. So this is the cost of ordering inventory u n minus 1. We also had a cost of holding a certain amount of inventory which is which was given by small r of x n minus 1 small r of x n minus 1. This was the cost of holding or a penalty for holding or a penalty for having unfulfilled demand plus now what one so this is the stage wise cost here this is the stage wise cost at stage n minus 1. This is the stage wise cost at stage n minus 1 and then you have a terminal cost which is which is capital R of x n. So you have a stage wise cost plus a capital R of x n. Now what does one do with this well let us analyze this further. So first let us look at the constraints. So there is a constraint on the action that u n minus 1 has to be from capital u n minus 1. Now what was capital u n minus 1? Capital u n minus 1 is the set of feasible actions or in equivalently the amount of feasible inventory that you can you can order you are allowed to order at stage n minus 1. Now at stage n minus 1 what you when your what you can the problem tells you that you can only order you can only order inventory you cannot give away or return inventory. So you can only order x additional stock you cannot return any any stock that you already have. So consequently u n minus capital u n minus 1 regardless of what the x n minus 1 is only says that the small u n minus 1 should be greater than equal to 0. In other words this this here this constraint here can simply be written in the following way it can be written as the minimum over u n minus 1 greater than equal to 0 of the expectation of these this term c n u n minus 1 plus small r of x n minus 1 plus capital r of x n all right. So this is this is where the this is what the the the expected cost is and you are trying to minimize this particular expected cost over all u n greater than equal to 0. Now let us let us think very carefully about each of these terms in the cost. What is what is a function of u n minus 1 here because since you have to minimize this whole thing this this this particular expression we need to ask what is the function of what what which of these is the function of u n minus 1. The other thing we need to ask is since we are taking an expectation here which of these quantities is actually random there are this this expression has 3 terms here but which of these is in fact random. Now if you so let us look at the second question first the in the the second question see the the quantity u n minus 1 is a constant that we want to choose it is a it is it is the it is the it is the it is a variable whose value we are choosing it is where it is it is a it is only a function of x n minus 1 okay. Now x n minus 1 is not the actually is not the realized state at time n minus 1 x n minus 1 was supposed to denote any any token state at time n minus 1. So this this particular when we are writing this each of these equations here the first in the first one we really mean to write this for all x n which means that this is the the x n here is just some is is merely any let us say representative way variable that denotes any value of this of the inventory at time n at time step n. Similarly here this is being written for all x n minus 1 as well okay. So if this is written for all x n minus 1 and this is written for all x n then what this is effectively telling us is that well we have this is what we have we have j n of x n equals r of x n x n for all x n and j n of x n minus 1 is equal to this for all x n minus 1 which which effectively means that x n minus 1 is really a parameter for as far as this optimization problem is concerned and as far as this expectation is concerned. So x n minus 1 is not the realized state and so therefore is consequently not random it is only a parameter as far as this expect this expectation and this optimization problem course. So consequently x n minus 1 is actually a is it is not random and as a result this expectation in this expectation this r of x n minus 1 can actually move out of the expectation. Now u n minus 1 is also a parameter as far as this expectation is concerned because u n minus 1 is to be is going to be realized is going to be chosen as a function of x n minus 1. It does not have access to the information that is of the demand that is going to be realized at time n in that during time period and during time period n minus 1. So all that all the randomness due to the demand at time n minus 1 is represented here in in x n it is what will manifest as different values of of x n. So the only quantity that is random here under this expectation is actually the x n.  So this is this is you can you can see here from from this that the randomness is all hidden in the hidden in the x n because after all the only the thing the thing that is random here is w n minus 1 which is the demand during time period n n minus 1. So x n minus 1 is fixed and deterministic u n minus 1 is is again a parameter as far as this expectation is concerned and it is it is going to be chosen as a function of x n minus 1 and w n minus 1 is the random demand at time n minus 1.  So this is therefore the therefore the expression it is minimum over u n minus 1 of this of this entire call. Now you look at now what is r of x n minus 1 well r of x n minus 1 here is a con is since x n minus 1 is a parameter r of x n minus 1 is actually not affected by this minimization this minimization takes r r it takes x n minus 1 as a parameter right.   And when one once if you if you know the an expression for r and if you know the value of c and and also for similarly for small r one can one can calculate this right. So this would then give us j n minus 1 of x n minus 1 for all x n minus 1 all right okay. So this is this is now the cost to go at times at stage n minus 1 okay. So what the dynamic programming algorithm now asks us to do the same thing for k equal to n minus 2 that means at stage n minus 2. So what we have done so far is we have looked at from if you go from time period 0, 1, 1, 2 and so on till till n and we solve the problem in this part of the tail of the time horizon starting from time period n minus 1 and ending at the beginning of time period n. Now what the algorithm asks us to do is to do the same thing starting from time period n minus 2 okay. So it is asking us to do the same thing for this for this time period. Now fortunately when we look at this longer time period which is starting from n minus 2 all the way till n this leg of it okay the n minus 1 to n leg of it has already been solved. We already know that regard what we already know from our earlier calculation what the cost to go was starting from stage n minus 1. So out here for instance so if what Jn minus 1 of xn minus 1 has told us is that regardless of which state we end up in okay it means regardless of what value xn minus 1 takes we now know what is going to be the optimal thing to do and the optimal cost starting from that state onwards till the end of the time horizon. So as a result of this starting from we know we already know that whatever state xn minus 1 we now end up in after taking some decisions at stage n minus 2 whatever state n minus 1 x n minus 1 we end up in we know what it is going what is the optimal cost from there onwards till till the end of the time horizon right. So as a result of that we can now compute what we can now write what the optimal cost is starting at any arbitrary state at time n minus 2. So Jn minus 2 which is the cost to go at state at time n minus 2 as a function of state xn minus 2 this should therefore be the minimum over all Un minus 2. Now Un minus 2 again is constrained to be greater than equal to 0 so this is greater than equal to 0 the expectation of the stage wise cost that you have in the stage wise cost that you have in state n minus 2 that stage wise cost is again given in terms of C times Un minus 2 plus R of xn minus 2 plus the optimal cost that you would incur if you end up in a state xn minus 1 at time n minus 1. So and that cost is simply the cost to go the cost to go at time n minus 1 right. So the Jn minus 1 expression that we just computed in the previous in the in the previous slide is to be plugged in is to be plugged in here. So that comes up here your stage wise cost comes up here. So this so what is this expression effectively telling us it is telling us you know the optimal thing to do when you are at stage n minus 2 is to look at the is to look at these two terms first is the stage wise cost that you will incur in this period plus the optimal thing that you will incur as a function of where you will land up after this after the first period. So the optimal thing you would do after you as a function of where you would land up where you would land up is your state xn minus 1 right. So that is that is given here the optimal the optimal cost that you will incur is given by Jn minus 1 of xn minus 1. So that the so you take a sum total of this this is the stage wise cost and the optimal cost to go you take the expectation of that and you minimize that overall overall u n minus u n minus 2 greater than equal to 0. This then will give us the the this then will give us the value function or the cost to go at stage n minus 2. We will do this calculation a little more in detail in the next part.