Question 1155269
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<pre>

Mathematical frame for this minimization problem is as it is presented in the post by @Theo

x = # of tables;
y = # of chairs;
Objective function P(x,y) = 20x + 16y
Constraints are

    3x + 2y <= 150
     x + 2y <=  75
     x >= 8, y >= 10.


   +--------------------------------------------------+
   |   B U T   B U T   B U T   B U T   B U T   B U T  |
   +--------------------------------------------------+


       But it is VERY specific / special maximization problem.


According to the context, it REQUIRES the solution x and y in INTEGER numbers.
Again : in integer numbers --- not in real.


So, it is Linear Programming problem <U>in integer numbers</U>.


It is very special/specific problem, and it requires ADEQUATE METHODS of solution.

Again : this very special/specific problem requires ADEQUATE METHODS of solution, different from that 
what are used in continuum minimization problems.


Look into the post by @Theo: you see there the grid of points in the feasibility quadrilateral.
Our task in this case is to find the maximal solution ON THIS GRID (!)


        ON THIS GRID only (!) --- it is the specific of this problem.


What you may find in the continuum model ---- IS NOT THE SOLUTION in integer numbers 
                                              and is not the solution to the problem as it is posed.


Now.  In the Internet, you may find online solvers for the continuum minimax problems.
      One of them is the solver used by @Theo. But, as I just explained, it is not an adequate tool in our specific case.


UNFORTUNATELY, in the Internet you will not find online (free of charge) solver for minimax problems with integer solutions.

               (at least, I did not find them, <U>really working</U>, although I spent hours for this search).


But I am a mathematician according to my basic education, and I was a computer programmer-analyst in my past life.

My life experience touched me finding an exit from <U>any complicated situation</U> to <U>REALLY SOLVE a problem</U>.


        This time I decided to use Excel to get the solution.

        The major idea was that the solution requires to analyze only FINITE set of grid points.


So, I created Excel spreadsheet and quickly got the solution.


This spreadsheet (the Table) is presented below.


First column (called x) is the number of tables. It goes from 8 (see constraint for x) to 44.
    (this magician number 44 came from @Theo solution),

Next column (called y1)     is  {{{y[1]}}} = {{{(150-3x)/2}}}.

Next column (called y1_int) is integer part of values of y1.

Next column (called y2)     is {{{y[2]}}} = {{{(75-x)/2}}}.

Next column (called y2_int) is integer part of values of y2.

Next column (called min(y1_int,y2_int) is for minimum ({{{y[1_int]}}},{{{y[2_int]}}}).

Finally, the last column is the Profit function, calculated on the grid.

As you should understand from my description, I consider the set of integer points of the grid,
belonging to the feasibility quadrilateral and  <U>CLOSEST</U>  to the boundary lines.



x	y1	y1_int	y2	y2_int	min		P=20x+16y		
					(y1_int,y2_int)			
8	63.0	63	33.5	33	33		688		
9	61.5	61	33.0	33	33		708		
10	60.0	60	32.5	32	32		712		
11	58.5	58	32.0	32	32		732		
12	57.0	57	31.5	31	31		736		
13	55.5	55	31.0	31	31		756		
14	54.0	54	30.5	30	30		760		
15	52.5	52	30.0	30	30		780		
16	51.0	51	29.5	29	29		784		
17	49.5	49	29.0	29	29		804		
18	48.0	48	28.5	28	28		808		
19	46.5	46	28.0	28	28		828		
20	45.0	45	27.5	27	27		832		
21	43.5	43	27.0	27	27		852		
22	42.0	42	26.5	26	26		856		
23	40.5	40	26.0	26	26		876		
24	39.0	39	25.5	25	25		880		
25	37.5	37	25.0	25	25		900		
26	36.0	36	24.5	24	24		904		
27	34.5	34	24.0	24	24		924		
28	33.0	33	23.5	23	23 		928		
29	31.5	31	23.0	23	23		948		
30	30.0	30	22.5	22	22		952		
31	28.5	28	22.0	22	22		972		
32	27.0	27	21.5	21	21		976		
33	25.5	25	21.0	21	21		996		
34	24.0	24	20.5	20	20		1000		
35	22.5	22	20.0	20	20		1020		
36	21.0	21	19.5	19	19		1024		
37	19.5	19	19.0	19	19		1044		
38	18.0	18	18.5	18	18		1048	(*)<<<---===	
39	16.5	16	18.0	18	16		1036		
40	15.0	15	17.5	17	15		1040		
41	13.5	13	17.0	17	13		1028		
42	12.0	12	16.5	16	12		1032		
43	10.5	10	16.0	16	10		1020		
44	9.0	9	15.5	15	9		1024		
				 				
							1048	<<<---=== MAX


Generating this spreadsheet is very easy: You simply move from on column to the other, from left to right.

Its creation was faster than your reading of my post, and MUCH more fast than my writing of this post.


When the last column is ready and is just filled with numbers, Excel allows to find the maximum in this column quickly.


This row with the solution is marked in my table by this sign (*)<<<---===.



So the problem is just solved, and the

<U>ANSWER</U> is  38 tables and 18 chairs with the maximum profit of 1048 dollars.
</pre>

The solution is <U>completed</U>


----------------


<U>The post-solution note</U>.


<pre>
    Eventually, the solution is close to the continuum solution, found by @Theo.

    But do not attach importance to this fact: 
                in integer mode solution, the optimum point can be far enough from the continuum solution.
</pre>


<U>Just &nbsp;&nbsp;D O N E</U>.