Each machine must be run by one of 19 cross-trained workers who are each available 35 hours per week. The plant has 10 type 1 machines available, 6 type 2 machines available, and 8 type 3 machines available. Products 1, 2, and 3 contribute $90, $120, and $150, respectively, in marginal profit per unit produced.

As from the given table product 1 uses 2 machine hours of machine 1, product 2 uses 2 machine hours of machine 1 and product 3 uses 1 hour of machine 1 so

[tex]2X+2Y+Z\leq 400[/tex]

Now as the number of hours in a week are 40 and there are a total of 6 type 2 machines so the total number of machine 2 hours are 40*6=240 hours

As from the given table product 1 uses 3 machine hours of machine 2, product 2 uses 4 machine hours of machine 2 and product 3 uses 6 hour of machine 2 so

[tex]3X+4Y+6Z\leq 240[/tex]

Now as the number of hours in a week are 40 and there are a total of 8 type 3 machines so the total number of machine 3 hours are 40*8=320 hours

As from the given table product 1 uses 4 machine hours of machine 3, product 2 uses 6 machine hours of machine 3 and product 3 uses 5 hour of machine 3 so

[tex]4X+6Y+5Z\leq 320[/tex]

Now as the machine 1 is used as 2X+2Y+Z in a week and the week is of 40 hours so the number of machines to be used are given as

[tex]\dfrac{2X+2Y+Z}{40}\leq 10[/tex]

Now as the machine 2 is used as 3X+4Y+6Z in a week and the week is of 40 hours so the number of machines to be used are given as

[tex]\dfrac{3X+4Y+6Z}{40}\leq 6[/tex]

Now as the machine 3 is used as 4X+6Y+5Z in a week and the week is of 40 hours so the number of machines to be used are given as

[tex]\dfrac{4X+6Y+5Z}{40}\leq 8[/tex]

Now the workers are available for 35 hours so the worker available at the machine 1 is given as

[tex]\dfrac{2X+2Y+Z}{35}[/tex]

That of machine 2 is given as

[tex]\dfrac{3X+4Y+6Z}{35}[/tex]

That of machine 3 is given as

[tex]\dfrac{4X+6Y+5Z}{35}[/tex]

As the total number of workers is 19 so the constraint is given as

[tex]\dfrac{2X+2Y+Z}{35}+\dfrac{3X+4Y+6Z}{35}+\dfrac{4X+6Y+5Z}{35}\leq 19[/tex]

So the Linear programming model is given as below

Profit Function: [tex]P=90X+120Y+150Z[/tex]

Constraints: