Is linear programming the double value of a variable of a simplex variable?

Question

I just learned the simplex method for solving linear programs, and I'm trying to figure out what the double problem is.

I understand the mechanics of solving a double problem - I do not need help with this. What I cannot get (even after reading about it on Wikipedia ) is the actual value of y variables in double.

I would like to give an example of everything together with variable values ​​in the main problem and what I understood from the dual, and I would ask anyone to kindly explain the values ​​in the double:

Primal:

max z = 3*x1 + 5*x2 subject to: x1 <= 4 2*x2 <= 12 3*x1 + 2*x2 <= 18 x1, x2 >= 0 

In the primary task x1 and x2 , the quantities of products A and B are presented. 3 and 5 - their sales prices, respectively. Products are manufactured on three M1-M3 machines. The first product requires an hour of work on M1 and 3 hours on M3. To get the second one requires two hours of work on M2 and M3. Machines M1, M2, M3 can operate for a maximum of 4, 12 and 18 hours, respectively. Finally, I cannot produce a negative amount of any of the products.

Now I ask a dual task:

 min z = 4*y1 + 12*y2 + 18*y3 subject to: y1 + 3*y3 >= 3 y2 + 2*y3 >= 5 y1, y2, y3 >= 0 

Now, the only thing that I think I can understand is that the restrictions mean: - for an hour of work on M1 and 3 hours on M3 I have to pay at least 3 currency units - for two hours of work on M2 and 2 hours on M3 I have to pay at least 5 units of money

But I just can not plunge into the meaning of the values ​​of the variables y1 and y2 . When I finally perform the minimization, the result in z is the same in the primary (although primary when increasing the lower bound of the result, while the double reduces the upper bound), but what does the object function of the double task consist of?

Answer 1

The function of your Dual's Objective is to minimize the Cost / Hour for the three machines (resources) .

So, the double objective function ( 4*y1 + 12*y2+ 18*y3 ) can be read as:

 Minimize 4*(cost/hour of Machine1) + 12*(cost/hour of M2) + 18*(cost/hr of M3) 

Since Primal deals with maximizing production profit, Dual can be seen as minimizing production costs for the firm.

(Sometimes it helps to think that the company “rents” M1, M2, and M3 cars.) If they are going to rent it, then what most of all should they pay [$ / hour] for each car and is it still profitable to produce x1 and x2 ?

The value of your double variables y1, y2, and y3 is the cost of ownership / rental per hour.

Parallel variables y are often called "shadow prices" of resources.

Since you are looking for an understanding of the understanding of duality :

• One trick is to reduce the dimension of the dual. (Imagine there was only one Machine M1.) Now formulate the double and try to understand the objective function and limitations.
• This helps to think in terms of “opportunity cost”. If a manufacturing company had to rent cars (resources), what price / hour should it pay? Alternatively, if there were many other (profitable) products, at what price / hour would machines be assigned to x1 and x2 instead of producing these other products.
• Please note that not all doubles are easily “understandable”. However, you can get an idea of ​​the many dual constraints by looking at the corresponding variable in the primary. Similarly, you can get an idea of ​​a double variable by examining the corresponding constraint.