MathCAD

       

Picture 3.11. The problem about paint


In Mathcad the report of  “ control weighing” you can look at the picture 3.11. Commentaries make clear what takes place in formulas. First of all the function Maximize gave fractional result as we had awaited  (look the point 2) –we can take fraction quantity of big drums so little drums are needless. Remembered epigraph and the title of the etude I had to go to excess of the versions. In Mathcad-document are formed two matrixes with name Îb (point 3.2) and St  (point 3.3). Their elements (there are 1088 elements, i.e. there are 17 columns and 64 lines) keep values of the volume (Îb) and of the worth (St) of paint depending on the combination of the packing. Then (point 3.4) some elements of the matrix Îb are named zero values if the given combinations of the parking do not suit by the worth. The rest is the sleight hand and no mathematics: in point 3.5 number of the line (variable N_15) and of the column (N_55) are determined of the matrix Îb. There is the element with maximum value on their crossing. The reply (6 little and 15 big drums) surprised Olga unpleasantly. She disappointed me unwittingly for 5 litres of paint and for 139 thousand roubles.

The method of searches the coordinates of maximum point that we can see at the picture 3.11 (the double sum) has the considerable limitation: one maximum element has to be in analyzable matrix (in our case it is matrix Ob). If there are a few elements than the reply will be wrong: the sums of the points’ coordinates with the maximum elements will be written to the variables N_15 and N_55. We have observed it in point 2 at the picture 3.9.

So Mathcad has spared me almost 150 thousand roubles. This money is not so great but if you added to them new computer etude, new theme of the lecture and new laboratory work and also the author’s emoluments for this book then the play was worth the candles.

Come back from Tambov to Moscow I analyzed this problem once more on my home computer in a comfortable atmosphere. And there is what I have got.


First of all if we change the initial installation of Solver the problem about paint can be right explained by the Solver of Excel. For it we could not be lazy and had to press the button Parameters… in the dialogue window Search of the

solution. In new dialogue window Parameters of the search of the solution

it was enough to decrease the admissible deviation from 5 to 1%. After it the right solution was found (15 big and 6 small drums). Frankly speaking in Excel Solver is not bad, but its initial installations are. There are few users Excel who apply Solver and press the button Parameters…

But who understand the heart of the problem of the optimization’s installations does not work with Excel. One gets some misunderstandings because of it.

In the second place both Olga and Excel, and Mathcad in different degree were disappointed me a little[10]: 910 litres of the paint could be packed another way-13 small and the same quality big drums[11]. In this case the balance was only 12 thousand roubles. And what is more the solution of the problem about paint with new criterion function (worth of paint in the drums) gives one more result: 37 small and 6 big drums. This way we get one thousand roubles else from Tambov-power.

Version of packing  (quantity of small drums/quantity of big drums)

2/16



6/15

13/13

37/6

Volume of paint (litres)

910

915

910

885

The rest of money (roubles)

186 000

47 000

12 000

11 000

In the third place when I showed this table in the supply department MPEI then I was told that the most optimum alternative both for me (money are important for me) and for MPEI (it needs paint) the fourth one: we get almost all money from Tambov-power. It seems strange but 885 litres of paint is more than 910 and 915. Point is that a lot of paint are lost because of tint from massive drums to small tare. The drum with 15 litres can be taken to the maintainable lecture hall and can be used completely.

The wrong solution is got not only because of bad methods or bad software, but because of user does not know exactly what he really wants. All programs of linear programming’s solutions require the clear formulation of only one criterion function[12]. The purpose is clear when we solve training problems. What purpose do we have for in our life? But it is the mathematics but it is the philosophy…



The excess method illustrated in the picture 3.11 has three own limitations:

1) the variables’ number can not be more than two as soon as in Mathcad there are vectors and matrixes but there are not tensors (three- and so dimensions matrixes)

2) when matrix is excessive dimension the computer refuses to deal with it and it makes a protest as «there is not enough memory»;

3) the calculation (if the excess can be called the calculation) can last to long i.e. for a long time.

Two limitations can be taken off when we go from the method of forming the matrix (the picture 3.11) to the method of excess of the versions. We remember only optimum plan that can be realized by the software. We have done it in our etude number 6 (look at the pictures 6.31 and 6.32).

The excess method turns out indispensable (that is it can not be so unnatural) when the problem is integer but it looses its linearity[13].  In this case traditional methods (for instance the simplex-method) often turn out feeble.

Nevertheless the excess method in Mathcad will be always the misinterpretation of the first water. Mathcad is the program of interpretive type with low speed of realization of initial text. For excess we need not only the compilers but also the compilers optimizing the time of realization the program.


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