Expert: Sherry Wallin - 11/24/2009

Question
(a) select appropriate variables;
(b) write the objective functions; (c) write the constraints as
inequalities.

Production Roberta Hernandez sells three items, A, B,
and C, in her gift shop. Each unit of A costs her $5 to buy,
$1 to sell, and $2 to deliver. For each unit of B, the costs
are $3, $2, and $1, respectively, and for each unit of C, the
costs are $6, $2, and $5, respectively. The profit on A is $4;
on B, $3; and on C, $3. How many of each should she
order to maximize her profit if she can spend $1200 on
buying costs, $800 on selling costs, and $500 on delivery
costs?

I know that x1, x2, and x3 are my variables for part A and I get that z = 4x1 + 3x2 + 3x3 is my objective function for B but I am confused on how to get part C which is the constraints.

PS - I apologize I cannot do the small 1's by the X's :)

Answer
Constraints are the boundaries of the problem. Like for instance the number of any item produced will have to be >= 0, which means x1>= 0, x2>=0, x3>=0.  Also her buying costs are constrained (bounded) by $1200 so 5x1+3x2+4x3<=1200; her selling costs are constrained by $800 so x1+2x2+3x3<=800; and finally her delivery costs are constrained by $500 so 2x1+x2+3x3<=500. Another note of importance, since she is in the business to make money you want her profit to exceed her costs, right? So you want the profit to be >= total costs. Her costs add up to $2500 so 4x1 + 3x2 + 3x3 >=2500

Math Prof