# What is the Objective Function in this problem?The cost to run Machine 1 for an hour is $2. During that hour, Machine 1 produces 240 bolts and 100 nuts. The cost to run Machine 2 for an hour is...

What is the Objective Function in this problem?

The cost to run Machine 1 for an hour is $2. During that hour, Machine 1 produces 240 bolts and 100 nuts. The cost to run Machine 2 for an hour is $2.40. During that hour, Machine 2 produces 160 bolts and 160 nuts. With a combined running time of no more than 30 hours, how long should each machine run to produce an order of at least 2080 bolts and 1520 nuts at the minimum operating cost?

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Machine 1 produces 240 bolts and 100 nuts in an hour and the cost to run it is $2 per hour. Machine 2 produces 160 bolts and 160 nuts in an hour and it running cost is $2.4 per hour. The order requires 2080 bolts and 1520 nuts to be produced.

The number of bolts required is 2080 - 1520 = 560 more than the number of nuts. Compared to Machine 2, Machine 1 produces 80 more bolts and 60 fewer nuts per hour.

Running Machine 1 alone requires 15.2 hours to produce enough number of both nuts and bolts for the order. The total cost of running the machine is $30.4. Machine 2 alone requires 13 hours to produce enough number of nuts and bolts for the order and the cost of operating it is $31.2

It is seen that using just one of the machines results in an excess production of either nuts or bolts. To correct this, both the machines should be used. Running Machine 1 for an hour compared to Machine 2 produces 80 more bolts and 60 fewer nuts. As the order requires 560 more bolts than nuts, it can be run for 4 hours. The remaining order is produced using Machine 2 in 7 hours. By doing this, the total number of bolts produced is 4*240 + 7*160 = 2080, the total number of nuts produced is 4*100 + 7*160 = 1520. The cost of running the machines in this case is 4*2 + 2.4*7 = $24.8

**The order is completed at a minimum operating cost of $24.8 by running Machine 1 for 4 hours and Machine 2 for 7 hours. **

The objective function is the function that we wish to minimize or maximize. Here it is the cost function.

The total cost is determined by the number of hours we run each machine.

Let C be the total cost, M1 the time in hours that we run machine 1, and M2 the time inhours we run machine 2.

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**Then the objective function is `C=2(M1)+2.4(M2)` **

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The constraints are :

`M1>=0,M2>=0`

`240(M1)+160(M2)>=2080`

`100(M1)+160(M2)>=1520`

`(M1)+(M2)<=30`

**Sources:**