Can anyone help with this problem?
You will be using the simplex method to find a solution for this "real life" scenario. While the length of this problem might be intimidating, you should find that the steps needed to solve it are laid out quite clearly.
STEP 1: Read through the following information carefully.
Electric companies generate power from a number of different energy sources. Bumpton Power Co. uses coal, oil, gas, and nuclear power for generation. Their goal is to maximize their profit, but a number of factors restrain their generation of power.
One constraint on the company is the limit on particulate pollution which they can release into the atmosphere. They are limited to 25600 lbs. per month of particulate pollution. Every gigawatt hour (gwh) of electricity generated by burning coal produces 100 lbs. of particulates. Oil produces 20 lbs. of particulates for each gwh of electricity generated, and natural gas produces 5 lbs. for each gwh. (Nuclear fuel does not involve combustion and therefore produces no particulates.)
The company also has restrictions placed on its sulphur dioxide (SO2) emissions. No more than 6400 lbs. of SO2 can be released into the atmosphere. The burning of coal releases 20 lbs. of SO2 for each gigawatt hour of electricity generated, burning oil produces 15 lbs. of SO2 for each gwh of electricity generated, and burning natural gas produces 3 lbs. of SO2 for each gwh of electricity generated. (Nuclear generation produces no SO2.)
Nuclear waste has no approved disposal mechanism. Because of this the company's use of nuclear energy is limited to production of no more than 240 gwh from nuclear energy.
Bumpton knows that it can sell no more than 640 gwh of electricity per month, so it limits its generation so as not to exceed that production level.
Coal and oil are hauled to the company site by train. Each gwh of power generated by burning coal requires 5 containers of coal, and each gwh of power generated from oil requires 2 containers of oil. The carrying capacity of the railroad track limits hauling to 1600 containers per month.
Generation of power using natural gas is limited by gas supply to no more than 160 gwh per month.
The company sells its electricity for $30,000 per gwh. However, this is by no means all profit. For each gwh of power generated by burning coal their fuel costs and operating costs are $12,000. For each gwh of power generated by burning oil their fuel costs and operating costs are $20,000. For each gwh of power generated by burning natural gas their fuel costs and operating costs are $24,000. And for each gwh of power generated by nuclear energy their fuel costs and operating costs are $26,000.
How many gwh of electricity should Bumpton Power Co. produce using each type of fuel in order to maximize their profit?
STEP 2: Your problem is to determine what fuel mix Bumpton Power Co. should use, and what the profit will be with that fuel mix. To do this, you'll need to first write your constraints and your profit function.
Use the following variable names:
x = # gwh produced by coal
y = # gwh produced by oil
z = # gwh produced by natural gas
w = # gwh produced by nuclear
There are six constraints (one for each bulleted item above). Create these six constraints, being careful to use the proper variables in each case.
The profit function is the most difficult part in setting up the problem. To set up your profit function, you'll need to use the following idea:
Profit = Selling price - Cost
So, for example, the profit from selling a gwh of energy produced from coal would be $30,000 - $12,000 = $18,000. Repeat this calculation for each of the four types of fuel to find the four values that you will use in your profit function. Create your profit function.
STEP 3: Rewrite your constraints using slack variables and create your matrix. Enter your matrix into the row operation tool.
Hints and Tips: Your matrix will be large, so using the row operation tool is essential. Many students find that using the fraction mode of the tool is helpful in this problem since the numbers you will be working with will not be whole numbers and can be quite difficult to read in decimal mode. (Note: If you've been using a hand-held calculator to work with matrices up to this point you will find that with a matrix this large it is impossible to see the entire matrix on the screen. This makes determining each pivot element very difficult and time consuming. Save time and frustration and use the row operation tool - you'll be glad you did.)
STEP 4: Using the simplex method, solve the problem and find the optimal solution. (Give your answers correct to the nearest whole gwh or to the nearest dollar, whichever applies.)
Hints and Tips: If you get some answers correct and some answers incorrect make sure that you've gotten rid of *all* the negative numbers in the bottom row. Additionally, there should be no negative numbers in the far right column at any point. Another common error that students have with this problem involves typos in the initial matrix. Check your initial matrix carefully before doing any pivoting to avoid frustration later.
In your optimal solution:
How many gwh of electricity are produced from burning coal?
How many gwh of electricity are produced from burning oil?
How many gwh of electricity are produced from burning natural gas?
How many gwh of electricity are produced from nuclear energy?
What is the maximum profit?
9/20/2011 5:56:14 PM