So that's trying to do all the analysis. Okay, someone says that we're going to have one more unit of woods. Let's see what's going to happen if we have one more unit of wood. So the worst case is that we may just change is six to seven. So on the graph six would become seven, and then we resolve the problem, all right? If we resolve the problem very quickly, we will see okay, so we're going to produce 3.5 tables in each day. And that's going to give us 10.5 as our new, optimal objective value. Okay, that, of course, is greater than nine in this particular case. So one unit of additional would actually is worth while to the level of $1.5, okay? So somehow that means if someone wants to offer as one additional unit of wood at $1 we should get it. We should purchase it because that's going to earn us 0.5 as the net benefit. Okay, that sounds good. On the other hand, if someone comes to you and say, hey, I have some labor, our here. You may purchase labor hours from me at $1 per hour, you may change your 6 to 7. But now in the second constraint, and then you may modify this graph by making the constraint 2 to its right hand side. So once we do that, we will see that okay, the optimal solution does not change, because what we are lack of is woods. We don't really need more labor hours, okay? So that makes sense. The new optimal objective value is still the same as the old one. If we purchase that particular resource labor hours, the net loss is actually $1. We don't really want to do that even if someone wants to give us for free. We don't want that. We don't need any more labor hours. So we may see that our attitude for these two resources obviously are different. If someone wants to give us more wood we are happy. If someone wants to give us more labor hours, we don't really care about that. So that directly goes to the definition for shadow prices. For each resource, there is a maximum amount of price that we are willing to pay for one additional unit. That depends on the net benefit of that one additional unit. For wood that one additional benefit is 1.5, okay? Giving me one more unit of wood, I would earn $1.5 more. So my maximum willingness to pay for that is 1.5. For labor hour that maximum willingness to pay is actually 0. So this motivate us to define shadow prices. So the definition actually, take some time. So thus read it carefully. Given a linear program that has an optimal solution, the shadow price of constraint is the amount of objective value increased, okay? Increased when the right hand side of that constraint is increased by 1 assuming what, the current optimal basis remain optimal. So there are several key words that I need to tell you in this particular definition. So first we're talking about the possibility for the right hand side of that constraint to be increased by 1. So when that happens, I want to ask what's going to happen to the amount of objective value that would be increased, okay? It doesn't really matter whether we are talking about a maximization problem or minimization problem. We only talk about how the objective value would be increased, okay? And finally, while that amount is defined to be the shadow price here, we assume that current optimal basis does not change, okay? So later we will give you one example showing you this. But at this moment, maybe from previous examples is easy to say okay, so for constraint 1, the shadow price is 1.5, okay? Because when I increase the right hand side of constraint 1 by 1, the objective value would be increased by 1.5. So that's why the shadow price for constraint 1 is 1.5. The shadow price for constraint 2 is 0. Because when you increase the right hand side of constraint 2 by 1, nothing really changed, okay? That's about Shadow Price. So now let's give you some ideas about what we mean by assuming the optimal basis does not change. So let's consider a slightly different example. In this example, our right hand side values are 4 and 4.5. So if we try to optimize this problem, then through some analysis, we would find okay an optimal solutions here 4,0 and we may earn 12 in total, okay? So now if we want to find a shadow price of constraint 1 typically, you would do this. You would change 4 to 5, okay? Change 4 to 5, because you increase the right hand side by 1. But once you do that, now you're going to resolve this problem, and you will see that the optimal solution now is binding at another set of constraints. Previously is the first constraint that is binding at an optimal solution. Now, once you alter the first constraint, what is binding at the new optimal solution is actually the second constraint. So when the set of binding constraint changes, the optimal basis also changes, okay? And then, in this case, the new optimal solution gives you 13.5 and then that goes to your new objective value. So you would see that the change is actually 1.5. But our definition of shadow prices, assuming the optimal basis, does not change. So what does that mean? That means we would assume the optimal basis is still there. We will still focus on the point that is having the two original constraint binding there, okay? We would assume that we are still talking about 5,0. If we do that, then this is going to give us 15. And then the shadow price is defined to be 3 instead of 1.5. So maybe you would ask why we want to define it in that way. So actually it is because shadow prices are considered as the rate of improvement. When we say, increase the right hand side by 1 blah, blah, blah, we are talking about actually the rate of improvement. So if increase that by 1 does not make sense or actually makes the optimal basis changes, how about 1.5? How about 1 over 5? How about 1 over 10? Somehow we are looking for the tiny change such that the optimal basis does not change. And under that condition, we're talking about the rate of improvement or the rate of increase, all right? So somehow that's the definition of shadow prices. We are assuming that the optimal basis does not change. So we are talking about the rate of improvement. We are not really talking about what's going to happen if the right hand side is increased by 1. Sometimes we say this, we say when the right hand side is increased by 1. But don't forget, we always focus on the case where the optimal solution, optimal basis is assumed to be the same. So the optimal solution obviously changes, but the optimal basis is assumed to be the same. We still consider the original optimal basis.
