Introduce four new binary variables, y_A, ..., y_D, whose individual value is one if that product can be sold, and zero otherwise. Require that y_A + y_B + y_C + y_D is less than or equal to 3 - hence you are not allowed to sell more than three different kinds of products. Also, to connect the sales volume to this binary variable, you need a linear inequality of the form

x_A

Dear Michael,

Thank you for your helpful answer. I use both Excel and Lingo to solve this, I write the code like this but it did not work! then I tried Matlab codes but there is a problem again!

in Lingo I wrote:

max=      7000*A*AA+44700*B*BB+12000*C*CC+48650*D*DD+26000*E*EE+42000*F*FF;

sub to: 1000*A+3000*B+500*C+3500*D1000*E+1000*F=0; B>=0; C>=0; D>=0; E>=0; F>=0;

but it says unbounded solution!

I wonder if you can help me.

Thank you in advance!

The objective function should only have terms like 

7000 times AA + 44700 times BB, and so on, 

and you need inequalities of the form

AA

Do not multiply A and AA - there is no need to, and adding such nonlinearities is a very bad thing to do. :-)

Thank you again

I don't understand why it is wrong! Please correct me if I am wrong: according to the first example:

I must maximize the income function so:

70*A*AA+150*B*BB+120*C*CC+160*D*DD

(A, B,... are the products and the numbers are incomes and AA , ... are the integer variables whose value is 0 if the product can't be sold!)

sub to:

10*A+30*B+20*C+40*D=0; B>=0; C>=0; D>=0; (all the variables must be positive)

And at first I tried this code without AA,... but it did not work!

thank you very much

Dear Michael,

could you please explain more about this constraint that you mentioned before?

"AA

You need to add one binary variable for each product, taking the value 1 if and only if a positive quantity of that product is bought.  Say the binary variables are AA, BB, CC and DD as above.  If you want only three types of product to be made, add the linear constraint AA + BB + CC + DD = 3.  But then you need additional constraints, preferably linear ones, linking the new binary variables with the original variables.  For example, you could add:

A

Dear Adam N. Letchford

Thank you a lot for your kind help. 

please correct me if I am wrong: 

If I add this constraint ((A

I don't see any need to modify the objective function.  In fact, if you modify it the way you have proposed, the function becomes nonlinear, which will cause problems for most optimisation software packages.

I recommend reading a textbook on mathematical programming, such as "Model Building in Mathematical Programming" by H.P. Williams.

Thanks a lot, I correct my model but there is something wrong again! because it does not omit one of the product as I want and there is an answer for all of the products!

max 70*A*AA+150*B*BB+120*C*CC+160*D*DD

A+B+C+D

The variables AA, ..., DD should be binary, not general integer.  And the objective function should not involve products of variables, as we said above.  just leave the objective function as it was originally.

Hello there,

I think that can be solved by The GAMS software..

that was my homework.a optimization in transport industry.