figuring out the constraints in a linear programing problem?

Question

im not looking for the answer, just help identifying the constraints would be much appreciated. i know i need to maximize z= 4x + 6y...but what the heck are my constraints? I feel like unnecessary extra information in this problem to trick me...

thanks so much for thet help!!!


Wooden Toys, Inc. manufactures two kinds of toy tops – a standard model and a deluxe model. To construct the tops, each standard model requires 2 hours lathe time and 2 hours finishing time, and each deluxe model requires 3 hours lathe time and 2 hours finishing time. The company has 3 lathe operators and 2 finishing workers, each of whom work 30 and 40 hours per week, respectively. Each standard top brings a profit of $4, and each deluxe model a profit of $6. Assuming all the tops made will be sold, how many of each should be made to maximize profit

Answer 1

not much extra information there



you have the correct profit equation

obviously with this equation the curve keeps going up if you produced 100 of the premium then you get 600 profit



so you need to figure out the amount of product that can be made in a week i think that is the information you are given for hours of work



I am not sure what the statement each of whom work 30 and 40 hours per week means you have 5 people there



who works 30 and who works 40



if the lathe operators work 30 then total hours available is 3 * 30 = 90



if the finish workers work 40 hours then the total amount is 2* 40 = 80



so now you know the total times that are available



we can produce product up to 90 hours of lathe time



that means if we only make premium then its



90/3 = 30 units



if we only produce standard model thats



90/2 = 45 units



you can see the profit from 30 deluxe is 180



profit from 45 standard is 180



but there is a second catch ...



there are only 80 hours of finish workers so that means



45*2 = 90 units we cant produce 45 units



we can only finish workers produce 40 units



thats 80 hours/2 = 40 units in total



so one of your constraints is 40 units

2x + 2y <= 80



x + y <= 40



i think that will give you a pretty good hint of how to proceed







x <= 40 - y

and you might want to use the constraint



2x + 3y <= 90



does that help give you some ideas

Answer 2

To construct x number of standard models and y number of deluxe models:



...you need 2x + 3y hours of lathe time. There are 3 lathes, which each work 30 hours per week, so 3(30) = 90 hours of total lathe time per week are available. Therefore, 2x + 3y ≤ 90.



...you need 2x + 2y hours of finishing time. There are 2 finishing workers, who each work 40 hours per week, so 2(40) = 80 hours of finishing time per week are available. Therefore, 2x + 2y ≤ 80.



Also, you cannot produce a negative number of standard models nor a negative number of deluxe models, so x ≥ 0 and y ≥ 0.



And those are your constraints.

Answer 3

let x = number of standard tops

let y = number of deluxe model



Maximize

z = 4x + 6y



subject to

x,y >=0



2x + 3y <= 3*30 (lathe time)

2x + 3y <= 90



2x + 2y <= 2*40 (finish time)

2x + 2y <= 80

x + y <= 40