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Linear and Network Optimization in Business!!!

Chapter 4, Case Problem 5: Duke Energy Coal Allocation
Duke Energy manufactures and distributes electricity to customers in the United States and Latin America. Duke purchased Cinergy Corporation, which has generating facilities and energy customers in Indiana, Kentucky, and Ohio. For these customers Cinergy has been spending $725 to $750 million each year for the fuel needed to operate its coal-fired and gas-fired power plants; 92% to 95% of the fuel used is coal. In this region, Duke Energy uses 10 coal-burning generating plants: five located inland and five located on the Ohio River. Some plants have more than one generating unit. Duke Energy uses 28–29 million tons of coal per year at a cost of approximately $2 million every day in this region.

The company purchases coal using fixed-tonnage or variable-tonnage contracts from mines in Indiana (49%), West Virginia (20%), Ohio (12%), Kentucky (11%), Illinois (5%), and Pennsylvania (3%). The company must purchase all of the coal contracted for on fixed-tonnage contracts, but on variable-tonnage contracts it can purchase varying amounts up to the limit specified in the contract. The coal is shipped from the mines to Duke Energy’s generating facilities in Ohio, Kentucky, and Indiana. The cost of coal varies from $19 to $35 per ton and transportation/delivery charges range from $1.50 to $5.00 per ton.

A model is used to determine the megawatt-hours (mWh) of electricity that each generating unit is expected to produce and to provide a measure of each generating unit’s efficiency, referred to as the heat rate. The heat rate is the total BTUs required to produce 1 kilowatt-hour (kWh) of electrical power.

Coal Allocation Model
Duke Energy uses a linear programming model, called the coal allocation model, to allocate coal to its generating facilities. The objective of the coal allocation model is to determine the lowest-cost method for purchasing and distributing coal to the generating units. The supply/availability of the coal is determined by the contracts with the various mines, and the demand for coal at the generating units is determined indirectly by the megawatt-hours of electricity each unit must produce.

The cost to process coal, called the add-on cost, depends on the characteristics of the coal (moisture content, ash content, BTU content, sulfur content, and grindability) and the efficiency of the generating unit. The add-on cost plus the transportation cost are added to the purchase cost of the coal to determine the total cost to purchase and use the coal.

Current Problem
Duke Energy signed three fixed-tonnage contracts and four variable-tonnage contracts. The company would like to determine the least-cost way to allocate the coal available through these contracts to five generating units. The relevant data for the three fixed-tonnage contracts are as follows:

For example, the contract signed with RAG requires Duke Energy to purchase 350,000 tons of coal at a price of $22 per ton; each pound of this particular coal provides 13,000 BTUs.

The data for the four variable-tonnage contracts follow:

For example, the contract with Consol, Inc., enables Duke Energy to purchase up to 200,000 tons of coal at a cost of $32 per ton; each pound of this coal provides 12,250 BTUs.

The number of megawatt-hours of electricity that each generating unit must produce and the heat rate provided are as follows:

For example, Miami Fort Unit 5 must produce 550,000 megawatt-hours of electricity, and 10,500 BTUs are needed to produce each kilowatt-hour.

The transportation cost and the add-on cost in dollars per ton are shown below.

Managerial Report
Prepare a report that summarizes your recommendations regarding Duke Energy’s coal allocation problem. Be sure to include information and analysis for the following issues:

1. Determine how much coal to purchase from each of the mining companies and how it should be allocated to the generating units. What is the cost to purchase, deliver, and process the coal?
   
   
2. Compute the average cost of coal in cents per million BTUs for each generating unit (a measure of the cost of fuel for the generating units).  What is the significance of the average cost in evaluating the efficiency of fuel consumption for different units?
   
   
3. Compute the average number of BTUs per pound of coal received at each generating unit (a measure of the energy efficiency of the coal received at each unit).
   
   
4. Suppose that Duke Energy can purchase an additional 80,000 tons of coal from American Coal Sales as an “all or nothing deal” for $30 per ton. Should Duke Energy purchase the additional 80,000 tons of coal?
   
   
5. Suppose that Duke Energy learns that the energy content of the coal from Cyprus Amax is actually 13,000 BTUs per pound. Should Duke Energy revise its procurement plan?
   
   
6. Duke Energy has learned from its trading group that Duke Energy can sell 50,000 megawatt-hours of electricity over the grid (to other electricity suppliers) at a price of $30 per megawatt-hour. Should Duke Energy sell the electricity? If so, which generating units should produce the additional electricity?
   
   
7. Miami Fort #5 is the oldest power plant in this region.  It generates more pollution than allowed by new federal regulations and must be updated to current standards in order to stay in operation.  Duke Energy has decided to decommission this plant.  If each remaining facility has the ability to increase its capacity by 25% through changes to their operation and maintenance practices, how should the 550,000mWh of Miami Fort #5’s expected output be planned across the remaining plants?
   
   
8. Before your recommendations are presented to the executive team, your manager has asked you to visit site to conduct an audit of the Heat Rate of each generating unit, since that is a critical value within your model.  You are also asked to visit 3 of the coal mines to verify the BTUs/lb testing data.  You will be arriving at the Cincinnati Airport.  Develop the shortest distance route to take you to each facility and back to the airport.  The addresses for each generating unit follows: 
   
   Using the information above, create the data table needed, and then state the order of visits and driving miles to make the circuit.