Question

A power company operates three power generation plants. One is a wind plant, and the other two consume a combination of Fuel 1 and Fuel 2, emitting carbon dioxide in the process. In addition, all three plants require maintenance. The amount of fuel consumed (in Mg), maintenance required (in person-hours), carbon dioxide (CO2) emitted (in Mg), and power generated (in MWh) per day of operation is as follows: Maintenance Fuel 1 Fuel 2 CO2 Power Plant Required Required Required Emitted Produced 1 20 0 0 0 20 2 13 10 15 12 32 3 18 30 40 29 40 Each MWh of power can be sold at £121 and there is no limit on the amount that can be sold. Over its next planning period, the company has 230 person-hours for maintenance, 75 Mg of Fuel 1, and 90 Mg of Fuel 2 available.

(a) Due to environmental regulations, they cannot emit more than 200Mg of CO2 in this period. The company wants to know how to operate its plants to generate as much revenue as possible (you may assume that there is no limit on the number of days a plant can operate in this period). Give a linear program that models this problem and state what each of your variables is meant to represent. You do not need to solve this program. [11]

(b) Suppose now that the company can emit more than 200Mg of CO2, but now loses £55 of revenue for each Mg emitted after the first 200Mg because it must purchase “CO2 credits”. The other resource constraints remain as stated. The company now wants to know how to operate its plants to generate as much revenue as possible (you may assume that there is no limit on the number of days that a plant can operate in this period). Give a linear program that models this problem. You do not need to solve this program.

Answer 1

GIVEN THAT :-

According to the question we have that :-a power genration compan runs three plants .

* one is totally of wind

** the other two consume a combination of fuel 1 and fuel 2

==> 230 persond for maintaince

20x1+13x2+18x3 is less than or equal to 230

==> 75 mg of fuel

10x2+30x3 is lessthan or equal to 75

==>90mg of fuel

15x2+40x3 is less than or equal to 90

so now to find the the following

TO FIND:-a) Give a linear program what each of your variable is meant to represent ?

let the x1,x2,x3 be the power generated in the plants 1,2,3.

==> maximum limit of co2 is 200 mg

12x2+29x3 is less than or equal to 200

for all the above eqations we get the linear programming model as Z=121(x1+x2+x3)

=====> 20x1+13x2+18x3<_230

10x2+30x3<_75

15x2+40x3<_90

12x2+29x3<_200

there fore

from all the above equations we have that

x1,x2,x3 >0

TO FIND :-b) Linear progamming for the model in the question .

now we have the total revenue as 55(12x2+29x3)

so the maximize Z=121(x1+x2+x3)-55(12x2+29x3)

so now

20x1+13x2+18x3<_230

10x2+30x3<_75

15x2+40x3<_90

12x2+29x3>_200

there fore

from all the above equations we have that

x1,x2,x3 >_0

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